Measures and applications of quantum correlations

We begin this section with a brief refresh of measurements in quantum mechanics [2, 32], in particular focusing on how local measurements can be used to define and motivate the notion of classically correlated states. A detailed introduction to measurements within the context of QCs can be found in [33].

A generalised quantum measurement can be mathematically described by resorting to the notion of a quantum instrument, which is a set $M=\{{\mu }_{i}\}$ of subnormalised (i.e., trace nonincreasing) completely positive maps ${\mu }_{i}$, that sum up to a completely positive trace preserving (CPTP) map, i.e., to a quantum channel $\mu := {\sum }_{i}{\mu }_{i}$. We can think of each subnormalised map ${\mu }_{i}$ as corresponding to a measurement outcome i. Given a state ρ, the probability of obtaining the outcome i when applying the quantum instrument $M$ to ρ is

$$\begin{eqnarray*}&&{p}_{i}={\rm{Tr}}\left((,{\mu }_{i}[\rho ],)\right),\end{eqnarray*}$$

and the corresponding state after the measurement, also called 'subselected' post-measurement state, is

$$\begin{eqnarray*}&&{\rho }_{i}={\mu }_{i}[\rho ]/{p}_{i}.\end{eqnarray*}$$

If the outcome of the measurement is not known, then the resultant state, simply called post-measurement state, is the statistical mixture

$$\begin{eqnarray}&&\displaystyle \sum _{i}\,{p}_{i}{\rho }_{i}=\displaystyle \sum _{i}\,{\mu }_{i}[\rho ]=: \mu [\rho ].\end{eqnarray} \tag{ 14 }$$

It is also useful to think about how the result of this quantum measurement is stored in a classical register. If we think of an orthonormal basis $\{\left|| ,i,\rangle \right.\rangle \}$ corresponding to the measurement outcomes $\{i\}$, the classical register will read $\left|| ,i,\rangle \right.\rangle \left.\langle \langle ,i,| \right|$ when the outcome i occurs. Furthermore, if the outcome is not known, then the system composed of the output quantum system and the output classical register will be overall in the state

$$\begin{eqnarray*}&&\displaystyle \sum _{i}\,{p}_{i}{\rho }_{i}\otimes \left|| ,i,\rangle \right.\rangle \left.\langle \langle ,i,| \right|=\displaystyle \sum _{i}{\mu }_{i}[\rho ]\otimes \left|| ,i,\rangle \right.\rangle \left.\langle \langle ,i,| \right|.\end{eqnarray*}$$

Consequently, by ignoring the measured quantum system, i.e., by tracing it out, we get the following classical output

$$\begin{eqnarray}&&\displaystyle \sum _{i}\,{p}_{i}\left|| ,i,\rangle \right.\rangle \left.\langle \langle ,i,| \right|=\displaystyle \sum _{i}{\rm{Tr}}\left((,{\mu }_{i}[\rho ],)\right)\left|| ,i,\rangle \right.\rangle \left.\langle \langle ,i,| \right|=: M[\rho ].\end{eqnarray} \tag{ 15 }$$

The map M transforming an arbitrary quantum state ρ into the classical state $M[\rho ]$ is referred to as the quantum-to-classical map (or measurement map) corresponding to the quantum instrument $M$ with subnormalised maps $\{{\mu }_{i}\}$.

We note that in order to know the probabilities p_i of each outcome i we do not need to know all the details of $M$. Conversely, we just need to know how the dual ${\mu }_{i}^{* }$ of every subnormalised map acts on the identity. Indeed,

$$\begin{eqnarray*}&&{p}_{i}={\rm{Tr}}\left((,{\mu }_{i}[\rho ],)\right)={\rm{Tr}}\left((,{\mu }_{i}[\rho ]{\mathbb{I}},)\right)={\rm{Tr}}\left((,\rho {\mu }_{i}^{* }[{\mathbb{I}}],)\right).\end{eqnarray*}$$

The operators ${M}_{i}:= {\mu }_{i}^{* }[{\mathbb{I}}]$ form a so-called positive operator valued measure (POVM), i.e., a set of positive semidefinite operators that sum up to the identity, ${\sum }_{i}{M}_{i}={\mathbb{I}}$.

On the other hand, we must stress that, in general, given only a POVM $\{{M}_{i}\}$ we can just determine the probabilities of the outcomes but we do not know how the quantum system transforms. For example, one can easily see that a given POVM $\{{M}_{i}\}$ is compatible with infinitely many quantum instruments, such as all the ones that can be defined as follows: ${\mu }_{i}[\rho ]={m}_{i}\rho {m}_{i}^{\dagger }$, where the m_i's are such that ${M}_{i}={m}_{i}^{\dagger }{m}_{i}$. Indeed, it is clear that such m_i's are not uniquely defined, but rather also any ${m}_{i}^{\prime }={U}_{i}{m}_{i}$, with U_i being arbitrary unitaries, are such that ${M}_{i}={{m}_{i}^{\prime }}^{\dagger }{m}_{i}^{\prime }$.

A special type of measurement is a projective measurement, for which the maps ${\mu }_{i}$ are Hermitian orthogonal projectors, i.e., they satisfy the constraint ${\mu }_{i}{\mu }_{j}={\delta }_{{ij}}{\mu }_{i}$. Furthermore, rank one measurements are those that consist only of rank one subnormalised maps ${\mu }_{i}$.

Let us now consider a quantum system composed of two subsystems A and B. A local generalised measurement (herein called an LGM) is a generalised measurement acting either only on A or only on B or separately on both A and B, and whose subnormalised maps are, respectively, either $\{{({\tilde{\pi }}_{A})}_{a}\otimes {{\mathbb{I}}}_{B}\}$ or $\{{{\mathbb{I}}}_{A}\otimes {({\tilde{\pi }}_{B})}_{b}\}$ or $\{{({\tilde{\pi }}_{A})}_{a}\otimes {({\tilde{\pi }}_{B})}_{b}\}$. We shall use a tilde from now on to denote an LGM, whereas the lack of a tilde will be reserved for the particular case of a local projective measurement (LPM).

The probability of obtaining the outcome a when performing an LGM only on A is

$$\begin{eqnarray*}&&{p}_{a}={\rm{Tr}}\left\{\{,\left((,{\left((,{\tilde{\pi }}_{A},)\right)}_{a}\otimes {{\mathbb{I}}}_{B},),[\right[{\rho }_{{AB}}],\}\right\},\end{eqnarray*}$$

with corresponding subselected post-measurement state

$$\begin{eqnarray*}&&{\rho }_{{AB}| a}=\left[[,\left((,{\left((,{\tilde{\pi }}_{A},)\right)}_{a}\otimes {{\mathbb{I}}}_{B},),[\right[{\rho }_{{AB}}],]\right]/{p}_{a}.\end{eqnarray*}$$

For each outcome a, one can associate a vector in an orthonormal basis $\{{\left|| ,a,\rangle \right.\rangle }_{{A}^{\prime }}\}$ representing a classical register $A^{\prime}$ (note that this basis may live in a different Hilbert space to that of subsystem A). By ignoring the state of the measured subsystem A, the resulting state of the classical register $A^{\prime}$ and of the unmeasured subsystem B is then $\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{{A}^{\prime }}\otimes {\rho }_{B| a}$, where ${\rho }_{B| a}={{\rm{Tr}}}_{A}({\rho }_{{AB}| a})$. Hence, if the result is not known, this composite system will be in the classical–quantum state

$$\begin{eqnarray}&&{\tilde{{\rm{\Pi }}}}_{A}[{\rho }_{{AB}}]:= \displaystyle \sum _{a}\,{p}_{a}\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{{A}^{\prime }}\otimes {\rho }_{B| a}.\end{eqnarray} \tag{ 16 }$$

Likewise, for an LGM on both subsystems A and B, the probability of getting respectively the outcomes a and b is

$$\begin{eqnarray*}&&{p}_{{ab}}={\rm{Tr}}\left\{\{,\left((,{\left((,{\tilde{\pi }}_{A},)\right)}_{a}\otimes {\left((,{\tilde{\pi }}_{B},)\right)}_{b},),[\right[{\rho }_{{AB}}],\}\right\}\end{eqnarray*}$$

with corresponding subselected post-measurement state

$$\begin{eqnarray*}&&{\rho }_{{AB}| {ab}}=\left[[,\left((,{\left((,{\tilde{\pi }}_{A},)\right)}_{a}\otimes {\left((,{\tilde{\pi }}_{B},)\right)}_{b},),[\right[{\rho }_{{AB}}],]\right]/{p}_{{ab}}.\end{eqnarray*}$$

Again, associating to each outcome a one vector of the orthonormal basis $\{{\left|| ,a,\rangle \right.\rangle }_{{A}^{\prime }}\}$ corresponding to a classical register $A^{\prime}$ and to each outcome b one vector of the orthonormal basis $\{{\left|| ,b,\rangle \right.\rangle }_{{B}^{\prime }}\}$ corresponding to a classical register $B^{\prime}$, the composite classical register reads $\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{{A}^{\prime }}\otimes \left|| ,b,\rangle \right.\rangle {\left.\langle \langle ,b,| \right|}_{{B}^{\prime }}$ when both outcomes a and b are recorded. By ignoring the state of the measured subsystems A and B, the resulting state of the classical registers $A^{\prime}$ and $B^{\prime}$ is then $\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{{A}^{\prime }}\otimes \left|| ,b,\rangle \right.\rangle {\left.\langle \langle ,b,| \right|}_{{B}^{\prime }}$. Hence, if the result of the LGM is unknown, the system composed of the two classical registers will be in the classical–classical state

$$\begin{eqnarray}&&{\tilde{{\rm{\Pi }}}}_{{AB}}[{\rho }_{{AB}}]:= \displaystyle \sum _{{ab}}\,{p}_{{ab}}\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{{A}^{\prime }}\otimes \left|| ,b,\rangle \right.\rangle {\left.\langle \langle ,b,| \right|}_{{B}^{\prime }}.\end{eqnarray} \tag{ 17 }$$

Analogous definitions can be obtained for LPMs by just removing the tilde. Notice that, according to Naimark's theorem, every generalised measurement can be represented as a projective measurement on a larger system [34]. In particular, this means that we can think of the action of an LGM applied only to subsystem A as equivalent to the action of an LPM applied only to a subsystem $A^{\prime}$ whose Hilbert space is obtained by extending the Hilbert space of A into a larger one. Similarly, we can think of the action of an LGM applied separately to both subsystems A and B as equivalent to the action of an LPM applied separately to both subsystem $A^{\prime}$ and $B^{\prime}$ whose Hilbert spaces are obtained by extending, respectively, the Hilbert spaces of A and B into larger ones. See e.g. [33] for a more detailed account.

Let us then consider in detail the relevant case of measuring ${\rho }_{{AB}}$ via a so-called complete rank one LPM, also known as a local von Neumann measurement, whose subnormalised maps form a complete set of orthogonal rank one projectors, i.e., ${({\pi }_{A})}_{a}[{\rho }_{A}]=\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{A}{\rho }_{A}\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{A}$ and ${({\pi }_{B})}_{b}[{\rho }_{B}]=\left|| ,b,\rangle \right.\rangle {\left.\langle \langle ,b,| \right|}_{B}{\rho }_{B}\left|| ,b,\rangle \right.\rangle {\left.\langle \langle ,b,| \right|}_{B}$, with $\{{\left|| ,a,\rangle \right.\rangle }_{A}\}$ and $\{{\left|| ,b,\rangle \right.\rangle }_{B}\}$ orthonormal bases and ${\rho }_{A}$ and ${\rho }_{B}$ arbitrary states for subsystem A and B respectively. Then, it is immediate to see that the post-measurement states can be written as

$$\begin{eqnarray}\begin{array}{rcl}{\pi }_{A}[{\rho }_{{AB}}] & := & \displaystyle \sum _{a}\left((,\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{A}\otimes {{\mathbb{I}}}_{B},)\right){\rho }_{{AB}}\left((,\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{A}\otimes {{\mathbb{I}}}_{B},)\right)\\ & = & \displaystyle \sum _{a}\,{p}_{a}\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{A}\otimes {\rho }_{B| a},\end{array}\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\pi }_{{AB}}[{\rho }_{{AB}}] & := & \displaystyle \sum _{{ab}}\left((,\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{A}\otimes \left|| ,b,\rangle \right.\rangle {\left.\langle \langle ,b,| \right|}_{B},)\right){\rho }_{{AB}}\left((,\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{A}\otimes \left|| ,b,\rangle \right.\rangle {\left.\langle \langle ,b,| \right|}_{B},)\right)\\ & = & \displaystyle \sum _{{ab}}\,{p}_{{ab}}\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{A}\otimes \left|| ,b,\rangle \right.\rangle {\left.\langle \langle ,b,| \right|}_{B},\end{array}\end{eqnarray} \tag{ 19 }$$

i.e., any state ${\rho }_{{AB}}$ (even if initially entangled or generally non-classical) is mapped into a classical–quantum or a classical–classical state, after such a complete rank one LPM acting on subsystem A or on both subsystems A and B, respectively. This simple observation captures the fundamental perturbing role of local (von Neumann) quantum measurements on general states of composite systems.

Indeed, since the action of any complete LPM ${\pi }_{A}$ on any input state ${\rho }_{{AB}}$ results in a classical–quantum post-measurement state of the form given in equation (18), if a state ${\rho }_{{AB}}$ is invariant under such an operation it must be necessarily classical–quantum. This means that the only states left invariant by a complete rank one LPM on subsystem A are classical–quantum states. Conversely, for every classical–quantum state ${\rho }_{{AB}}\in {C}_{A}$ as defined in equation (6), there exists a complete rank one LPM on subsystem A which leaves such a state invariant, defined precisely as the one with a complete set of orthogonal rank one projectors given by ${({\pi }_{A})}_{i}[{\rho }_{A}]=\left|| ,i,\rangle \right.\rangle {\left.\langle \langle ,i,| \right|}_{A}{\rho }_{A}\left|| ,i,\rangle \right.\rangle {\left.\langle \langle ,i,| \right|}_{A}$. It thus holds that

$$\begin{eqnarray}&&{\rho }_{{AB}}\in {C}_{A}\quad \iff \quad \exists {\rm{}}\,{\rm{a}}\,{\rm{complete}}\,{\rm{rank}}\,{\rm{one}}\,{\rm{LPM}}\,{\rm{on}}\ A{\rm{}}\,{\rm{such}}\,{\rm{that}}\ {\pi }_{A}[{\rho }_{{AB}}]={\rho }_{{AB}}.\end{eqnarray} \tag{ 20 }$$

A similar argument for classical–classical states and LPMs on both subsystems implies that

$$\begin{eqnarray}&&{\rho }_{{AB}}\in {C}_{{AB}}\quad \iff \quad \exists {\rm{}}\,{\rm{a}}\,{\rm{complete}}\,{\rm{rank}}\,{\rm{one}}\,{\rm{LPM}}\,{\rm{on}}\ A\ {\rm{and}}\ B\ {\rm{such}}\,{\rm{that}}\ {\pi }_{{AB}}[{\rho }_{{AB}}]={\rho }_{{AB}}.\end{eqnarray} \tag{ 21 }$$

These two statements formalise the defining property 2 given earlier for classical states, and illustrated in figure 3(a). Moreover, they also motivate why they are called classically correlated states: because it is possible to perform a local von Neumann measurement on them that leaves such states unperturbed.

Even more, since the state of a measured subsystem becomes itself diagonal in the basis in which the complete rank one LPM is performed, we have that the corresponding post-measurement state ${\pi }_{A}[{\rho }_{{AB}}]$ (resp., ${\pi }_{{AB}}[{\rho }_{{AB}}]$) and output of the quantum-to-classical map ${{\rm{\Pi }}}_{A}[{\rho }_{{AB}}]$ (resp., ${{\rm{\Pi }}}_{{AB}}[{\rho }_{{AB}}]$) can be considered equivalent.

Overall, we can thus locally access the information about a system in a classical state by probing the subsystem via a von Neumann measurement without inducing a global disturbance (in contrast to the general quantum mechanical scenario). This latter feature reveals that the correlations that are left after a complete rank one LPM are akin to those described by classical probability theory, hence justifying the terminology for the corresponding output states.

Let us proceed by analysing in more detail the workings of an LPM according to von Neumann's model [32]. Given a bipartite system in the initial state ${\rho }_{{AB}}$, any LPM on subsystem A with a complete set of orthogonal rank one projectors ${({\pi }_{A})}_{a}[{\rho }_{A}]=\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{A}{\rho }_{A}\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{A}$ can be realised by letting subsystem A interact via a unitary operation ${V}_{{AA}^{\prime} }^{\{\left|| ,a,\rangle \right.\rangle \}}$ with an ancillary system $A^{\prime}$ initialised in a reference pure state, say ${\left|| ,0,\rangle \right.\rangle }_{{A}^{\prime }}$ in its computational basis; in this case the ancillary system $A^{\prime}$ has the same dimension as A and plays the role of a measurement apparatus [35]. The state of the three parties $A,B,A^{\prime}$ after the interaction is known as the pre-measurement state

$$\begin{eqnarray}&&\rho {^{\prime} }_{{ABA}^{\prime} }^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}:= ({V}_{{AA}^{\prime} }^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}\otimes {{\mathbb{I}}}_{B})({\rho }_{{AB}}\otimes \left|| ,0,\rangle \right.\rangle {\left.\langle \langle ,0,| \right|}_{{A}^{\prime }}){\left((,{V}_{{AA}^{\prime} }^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}\otimes {{\mathbb{I}}}_{B},)\right)}^{\dagger }.\end{eqnarray} \tag{ 22 }$$

The LPM is completed by a readout of the apparatus $A^{\prime}$ in its eigenbasis, in such a way that

$$\begin{eqnarray}&&{{\rm{Tr}}}_{{A}^{\prime }}\rho {^{\prime} }_{{ABA}^{\prime} }^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}=\displaystyle \sum _{a}\left((,\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{A}\otimes {{\mathbb{I}}}_{B},)\right){\rho }_{{AB}}\left((,\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{A}\otimes {{\mathbb{I}}}_{B},)\right)={\pi }_{A}[{\rho }_{{AB}}].\end{eqnarray} \tag{ 23 }$$

Imposing equation (23), one finds that the pre-measurement interaction between A and $A^{\prime}$ has to be of a very specific form (up to a local unitary on $A^{\prime}$), namely that of an isometry

$$\begin{eqnarray}&&{V}_{{AA}^{\prime} }^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}{\left|| ,a,\rangle \right.\rangle }_{A}{\left|| ,0,\rangle \right.\rangle }_{{A}^{\prime }}={\left|| ,a,\rangle \right.\rangle }_{A}{\left|| ,a,\rangle \right.\rangle }_{{A}^{\prime }}.\end{eqnarray} \tag{ 24 }$$

We can always think of ${V}_{{AA}^{\prime} }^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}$ as the combination ${V}_{{AA}^{\prime} }^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}={C}_{{AA}^{\prime} }({U}_{A}^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}\otimes {{\mathbb{I}}}_{{A}^{\prime }})$ of a local unitary ${U}_{A}^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}$ on A, which serves the purpose of determining in which basis $\{{\left|| ,a,\rangle \right.\rangle }_{A}\}$ the subsystem is going to be measured (i.e., which observable is being probed), followed by a generalised controlled-not gate ${C}_{{AA}^{\prime} }$, whose action on the computational basis ${\left|| ,i,\rangle \right.\rangle }_{A}{\left|| ,i^{\prime} ,\rangle \right.\rangle }_{{A}^{\prime }}$ of ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d}$ is ${C}_{{AA}^{\prime} }{\left|| ,i,\rangle \right.\rangle }_{A}{\left|| ,i^{\prime} ,\rangle \right.\rangle }_{{A}^{\prime }}={\left|| ,i,\rangle \right.\rangle }_{A}{\left|| ,i\oplus i^{\prime} ,\rangle \right.\rangle }_{{A}^{\prime }}$, with ⊕ denoting addition modulo d [28, 36].

In case of a joint LPM on both subsystems A and B, the pre-measurement state reads

$$\begin{eqnarray}&&\rho {^{\prime} }_{{ABA}^{\prime} B^{\prime} }^{\{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}}:= ({V}_{{AA}^{\prime} }^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}\otimes {V}_{{BB}^{\prime} }^{\{{\left|| ,b,\rangle \right.\rangle }_{B}\}})({\rho }_{{AB}}\otimes \left|| ,0,\rangle \right.\rangle {\left.\langle \langle ,0,| \right|}_{{A}^{\prime }}\otimes \left|| ,0,\rangle \right.\rangle {\left.\langle \langle ,0,| \right|}_{{B}^{\prime }}){\left((,{V}_{{AA}^{\prime} }^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}\otimes {V}_{{BB}^{\prime} }^{\{{\left|| ,b,\rangle \right.\rangle }_{B}\}},)\right)}^{\dagger },\end{eqnarray} \tag{ 25 }$$

where $B^{\prime}$ is an additional ancilla of the same dimension as B, which plays the role of an apparatus measuring B in the basis $\{{\left|| ,b,\rangle \right.\rangle }_{B}\}$.

In general, the pre-measurement state $\rho {^{\prime} }_{{ABA}^{\prime} }^{\{\left|| ,a,\rangle \right.\rangle \}}$ ($\rho {^{\prime} }_{{ABA}^{\prime} B^{\prime} }^{\{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}}$) is entangled across the split ${AB}\,:A^{\prime}$ (${AB}:A^{\prime} B^{\prime}$), meaning that the ancilla(e) become entangled with the whole system AB due to the pre-measurement interaction(s); it is indeed because of the presence of such entanglement that the system AB is mapped to a classical state upon tracing over the ancillary system(s), an act which generally amounts to an irreversible information loss similar to a decoherence phenomenon, perceived as a disturbance on the state of the system due to the local measurement(s) [37]. However, we know that classical states can be locally measured without perturbation. In the present context, this means that for classical states there is at least one local measurement basis such that the corresponding pre-measurement state remains separable across the system versus ancilla(e) split. Formally, it has been proven in [28, 35, 36] that

$$\begin{eqnarray}&&{\rho }_{{AB}}\in {C}_{A}\,\,\,\,\iff \,\,\,\,\exists \text{a local basis}\,\,\{{\left|| ,a,\rangle \right.\rangle }_{A}\}\,\,{\rm{such}}\,{\rm{that}}\,\,\rho {^{\prime} }_{{ABA}^{\prime} }^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}\in {S}_{{AB}:A^{\prime} },\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&{\rho }_{{AB}}\in {C}_{{AB}}\,\,\,\,\iff \,\,\,\,\exists \text{local bases}\,\,\{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}\,\,{\rm{such}}\,{\rm{that}}\,\,\rho {^{\prime} }_{{ABA}^{\prime} B^{\prime} }^{\{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}}\in {S}_{{AB}:A^{\prime} B^{\prime} },\end{eqnarray} \tag{ 27 }$$

where we denote by ${S}_{X:Y}$ the set of separable states, equation (3), with the explicit indication of the considered bipartition $X\,:Y$. These two statements formalise the defining property 3 of classical states, as illustrated in figure 3(b). Equivalently, one may say that classical correlations are not always activated into entanglement by the act of local measurements [36, 38].

The possibility of invariance under a non-trivial local unitary evolution is another distinctive characteristic of classical states. Consider a local unitary ${U}_{A}^{{\rm{\Gamma }}}$ with a spectrum Γ that acts on subsystem A of a bipartite system in the state ${\rho }_{{AB}}$. With a slight abuse of notation, we can define the transformed state after the action of such local unitary as ${U}_{A}^{{\rm{\Gamma }}}[{\rho }_{{AB}}]$, with

$$\begin{eqnarray}&&{U}_{A}^{{\rm{\Gamma }}}[{\rho }_{{AB}}]:= \left((,{U}_{A}^{{\rm{\Gamma }}}\otimes {{\mathbb{I}}}_{B},)\right){\rho }_{{AB}}{\left((,{U}_{A}^{{\rm{\Gamma }}}\otimes {{\mathbb{I}}}_{B},)\right)}^{\dagger }.\end{eqnarray} \tag{ 28 }$$

In general, if we fix a non-degenerate spectrum Γ, the transformed state ${U}_{A}^{{\rm{\Gamma }}}[{\rho }_{{AB}}]$ will be different from the initial ${\rho }_{{AB}}$. However, if, and only if, ${\rho }_{{AB}}$ is classical–quantum, then one can find at least one local non-degenerate unitary which leaves this state invariant. Clearly, such a unitary ${U}_{A}^{{\rm{\Gamma }}}$ is characterised by having its eigenbasis given precisely by the orthonormal basis $\{{\left|| ,i,\rangle \right.\rangle }_{A}\}$ entering the definition of the classical–quantum state ${\rho }_{{AB}}$ as in equation (6). Formally, it thus holds that [39–41]

$$\begin{eqnarray}&&{\rho }_{{AB}}\in {C}_{A}\quad \iff \quad \exists \text{a local unitary}\,{U}_{A}^{{\rm{\Gamma }}}\ {\rm{with}}\,{\rm{non}} \mbox{-} {\rm{degenerate}}\,{\rm{\Gamma }}\,{\rm{such}}\,{\rm{that}}\,{\rm{}}{U}_{A}^{{\rm{\Gamma }}}[{\rho }_{{AB}}]={\rho }_{{AB}},\end{eqnarray} \tag{ 29 }$$

which formalises the defining property 4 of classical states, as illustrated in figure 3(c).

The spectrum Γ must be non-degenerate to exclude such trivialities as the choice of ${{\mathbb{I}}}_{A}$ as a local unitary (which would leave any state, not only those classical–quantum, invariant). Notice that, in contrast with the case of a local measurement discussed in section 2.3.1, a local unitary does not alter the correlations between A and B at all, yet can assess their nature based on the global change (or lack thereof) of the state ${\rho }_{{AB}}$ in response to the action of any such local unitary [41].

Curiously, unlike all the other definitions discussed in the current section, a necessary and sufficient condition analogous to equation (29) does not hold for classical–classical states in this case. Namely, considering also a local unitary ${U}_{B}^{{\rm{\Gamma }}}$ acting on B with the same spectrum Γ, and defining the joint action of ${U}_{A}^{{\rm{\Gamma }}}$ and ${U}_{B}^{{\rm{\Gamma }}}$ on ${\rho }_{{AB}}$ as

$$\begin{eqnarray}&&{U}_{{AB}}^{{\rm{\Gamma }}}[{\rho }_{{AB}}]:= \left((,{U}_{A}^{{\rm{\Gamma }}}\otimes {U}_{B}^{{\rm{\Gamma }}},)\right){\rho }_{{AB}}{\left((,{U}_{A}^{{\rm{\Gamma }}}\otimes {U}_{B}^{{\rm{\Gamma }}},)\right)}^{\dagger },\end{eqnarray} \tag{ 30 }$$

then if Γ is non-degenerate it still holds that

$$\begin{eqnarray}&&{\rho }_{{AB}}\in {C}_{{AB}}\Rightarrow \exists \text{local unitaries}\ {U}_{A}^{{\rm{\Gamma }}},{U}_{B}^{{\rm{\Gamma }}}{\rm{with}}\,{\rm{non}} \mbox{-} {\rm{degenerate}}\ {\rm{\Gamma }}\ {\rm{such}}\,{\rm{that}}\ {U}_{{AB}}^{{\rm{\Gamma }}}[{\rho }_{{AB}}]={\rho }_{{AB}},\end{eqnarray} \tag{ 31 }$$

but the reverse implication is no longer true. In fact, there are even entangled states ${\rho }_{{AB}}$ which may be left unchanged by a joint non-trivial local unitary evolution; an example is the maximally entangled two-qubit Bell state ${\left|| ,{\rm{\Phi }},\rangle \right.\rangle }_{{AB}}=\tfrac{1}{\sqrt{2}}({\left|| ,0,\rangle \right.\rangle }_{A}{\left|| ,0,\rangle \right.\rangle }_{B}+{\left|| ,1,\rangle \right.\rangle }_{A}{\left|| ,1,\rangle \right.\rangle }_{B})$, which is invariant under the action of any tensor product of two matching Pauli matrices,

$$\begin{eqnarray}&&({{\sigma }_{i}}_{A}\otimes {{\sigma }_{i}}_{B})\left|| ,{\rm{\Phi }},\rangle \right.\rangle {\left.\langle \langle ,{\rm{\Phi }},| \right|}_{{AB}}({{\sigma }_{i}}_{A}\otimes {{\sigma }_{i}}_{B})=\left|| ,{\rm{\Phi }},\rangle \right.\rangle {\left.\langle \langle ,{\rm{\Phi }},| \right|}_{{AB}},\end{eqnarray} \tag{ 32 }$$

for $i=1,2,3$ (notice that the Pauli matrices are unitary and Hermitian and with non-degenerate spectrum ${\rm{\Gamma }}=\{\pm 1\})$. It is an open question how to give a suitable local unitary based condition characterising the set of classical–classical states, with one possibility being to fix a different spectrum between A and B, or to involve multiple local unitaries.

The previous analysis reveals that classical–quantum states commute with at least a local unitary ${U}_{A}^{{\rm{\Gamma }}}$ on subsystem A. An alternative way of describing this is by saying that classical–quantum states are locally incoherent with respect to the eigenbasis of ${U}_{A}^{{\rm{\Gamma }}}$. To be more precise, let us recall some basic terminology used in the characterisation of quantum coherence, see e.g. [42–45].

Quantum coherence represents superposition with respect to a fixed reference orthonormal basis. In the case of a single system, fixing a reference basis $\{\left|| ,i,\rangle \right.\rangle \}$, the corresponding set ${I}^{\{\left|| ,i,\rangle \right.\rangle \}}$ of incoherent states (i.e., states with vanishing coherence) is defined as the set of those states diagonal in the reference basis,

$$\begin{eqnarray}&&{I}^{\{\left|| ,i,\rangle \right.\rangle \}}:= \left\{\rho \quad | \quad \rho =\displaystyle \sum _{i}\,{p}_{i}\left|| ,i,\rangle \right.\rangle \left.\langle \langle ,i,| \right|\right\},\end{eqnarray} \tag{ 33 }$$

with $\{{p}_{i}\}$ a probability distribution. Any other state $\rho \,\not\hspace{-2pt}{\in }\,{I}^{\{\left|| ,i,\rangle \right.\rangle \}}$ exhibits quantum coherence with respect to the fixed reference basis $\{\left|| ,i,\rangle \right.\rangle \}$.

Consider now a bipartite system AB. We can fix local reference bases $\{{\left|| ,a,\rangle \right.\rangle }_{A}\}$ for subsystem A and $\{{\left|| ,b,\rangle \right.\rangle }_{B}\}$ for subsystem B, and characterise coherence with respect to these. In particular, we can define two different sets of locally incoherent states. Namely, the set ${I}_{A}^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}$ of incoherent-quantum (or incoherent on A) states is defined as [46, 47]

$$\begin{eqnarray}&&{I}_{A}^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}:= \left\{{\rho }_{{AB}}\quad | \quad {\rho }_{{AB}}=\displaystyle \sum _{a}\,{p}_{a}\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{A}\otimes {\rho }_{B}^{(a)}\right\},\end{eqnarray} \tag{ 34 }$$

for any quantum states $\{{\rho }_{B}^{(a)}\}$ of subsystem B, with $\{{p}_{a}\}$ a probability distribution. Similarly, the set ${I}_{{AB}}^{\{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}}$ of incoherent-incoherent (or incoherent on A and B) states is defined as [44, 48, 49]

$$\begin{eqnarray}&&{I}_{{AB}}^{\{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}}:= \left\{{\rho }_{{AB}}\quad | \quad {\rho }_{{AB}}=\displaystyle \sum _{{ab}}\,{p}_{{ab}}\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{A}\otimes \left|| ,b,\rangle \right.\rangle {\left.\langle \langle ,b,| \right|}_{B}\right\},\end{eqnarray} \tag{ 35 }$$

with $\{{p}_{{ab}}\}$ a joint probability distribution. Notice that in the above definitions the reference bases are fixed, which makes equations (34) and (35) different from the definitions of the sets of classical–quantum and classical–classical states, equations (7) and (11), respectively. However, it is straightforward to realise that a state ${\rho }_{{AB}}$ is classical–quantum (classical–classical) if, and only if, there exist a local basis for A (and for B) with respect to which ${\rho }_{{AB}}$ is incoherent on A (and B). In formulae,

$$\begin{eqnarray}&&{\rho }_{{AB}}\in {C}_{A}\quad \iff \quad \exists \text{a local basis}\,\,\{{\left|| ,a,\rangle \right.\rangle }_{A}\}\,\,{\rm{such}}\,{\rm{that}}\,\,{\rho }_{{AB}}\in {I}_{A}^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}},\end{eqnarray} \tag{ 36 }$$

$$\begin{eqnarray}&&{\rho }_{{AB}}\in {C}_{{AB}}\quad \iff \quad \exists \text{local bases}\,\,\{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}\,\,{\rm{such}}\,{\rm{that}}\,\,{\rho }_{{AB}}\in {I}_{{AB}}^{\{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}}.\end{eqnarray} \tag{ 37 }$$

These characterisations amount to the defining property 5 of classical states, as illustrated in figure 3(d). In the present paradigm, classicality is thus pinned down as a very intuitive notion, intimately related to the lack of quantum superposition in a specific local reference frame.

Finally, a classical state can be identified as a fixed point of an entanglement-breaking channel [50]. Entanglement-breaking channels [51] acting on subsystem A of a bipartite system are defined as those CPTP operations ${{\rm{\Lambda }}}_{A}^{\mathrm{EB}}$ which map any state ${\rho }_{{AB}}$ to a separable state, i.e.,

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{A}^{\mathrm{EB}}[{\rho }_{{AB}}]:= ({{\rm{\Lambda }}}_{A}^{\mathrm{EB}}\otimes {{\mathbb{I}}}_{B})[{\rho }_{{AB}}]\in {S}_{{AB}}\,\ \ \forall {\rho }_{{AB}}\in {D}_{{AB}}.\end{eqnarray} \tag{ 38 }$$

Equivalently, the entanglement-breaking channels are all and only the maps that can be written as the concatenation of a local measurement followed by a preparation [51], i.e.,

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{A}^{\mathrm{EB}}[{\rho }_{A}]:= \displaystyle \sum _{i}\,{\rm{Tr}}({M}_{i}{\rho }_{A}){\rho }_{A}^{(i)},\end{eqnarray} \tag{ 39 }$$

where the $\{{M}_{i}\}$ constitute a POVM while the $\{{\rho }_{A}^{(i)}\}$ are arbitrary quantum states of subsystem A. In other words, an entanglement-breaking channel applied to the subsystem state ${\rho }_{A}$ can be simulated classically as follows: a sender performs an LGM with POVM elements $\{{M}_{i}\}$ on subsystem A and sends the outcome i, which occurs with probability ${p}_{i}={\rm{Tr}}({M}_{i}{\rho }_{A})$, via a classical channel to a receiver, who then prepares the subsystem A in a corresponding pre-arranged state ${\rho }_{A}^{(i)}$ [51].

It then turns out that a state ${\rho }_{{AB}}$ is classical–quantum if, and only if, it is left invariant by at least one entanglement-breaking channel acting on subsystem A [50], i.e.,

$$\begin{eqnarray}&&{\rho }_{{AB}}\in {C}_{A}\quad \iff \quad \exists \text{an entanglement-breaking channel}\ \ {{\rm{\Lambda }}}_{A}^{\mathrm{EB}}\ {\rm{such}}\,{\rm{that}}\ {{\rm{\Lambda }}}_{A}^{\mathrm{EB}}[{\rho }_{{AB}}]={\rho }_{{AB}}.\end{eqnarray} \tag{ 40 }$$

To see this, let us first assume that ${\rho }_{{AB}}\in {C}_{A}$, i.e., ${\rho }_{{AB}}$ is defined as in equation (6); we have then from section 2.3.1 that such a state is invariant under the complete LPM with rank one projectors $\{\left|| ,i,\rangle \right.\rangle {\left.\langle \langle ,i,| \right|}_{A}\}$, which is a particular case of an entanglement-breaking channel. On the other hand, if there exists at least one entanglement-breaking channel ${{\rm{\Lambda }}}_{A}^{\mathrm{EB}}$ such that ${{\rm{\Lambda }}}_{A}^{\mathrm{EB}}[{\rho }_{{AB}}]={\rho }_{{AB}}$, then also the channel ${\overline{{\rm{\Lambda }}}}_{A}^{\mathrm{EB}}$, defined as ${\overline{{\rm{\Lambda }}}}_{A}^{\mathrm{EB}}={\mathrm{lim}}_{N\to \infty }\,\tfrac{1}{N}{\sum }_{n=1}^{N}{\left((,{{\rm{\Lambda }}}_{A}^{\mathrm{EB}},)\right)}^{n}$, leaves ${\rho }_{{AB}}$ invariant. In [52] it has been shown that this map is an entanglement-breaking channel as well, whose action can be written as follows, ${\overline{{\rm{\Lambda }}}}_{A}^{\mathrm{EB}}[{\rho }_{A}]={\sum }_{i}{\rm{Tr}}({M}_{i}{\rho }_{A}){\sigma }_{A}^{(i)}$, where the $\{{M}_{i}\}$ form a POVM while the $\{{\sigma }_{A}^{(i)}\}$ are states of subsystem A with orthogonal support. Therefore the channel ${\overline{{\rm{\Lambda }}}}_{A}^{\mathrm{EB}}$ transforms ${\rho }_{{AB}}$ into a classical–quantum state, and since ${\rho }_{{AB}}$ is invariant with respect to the action of this channel, then ${\rho }_{{AB}}$ must be classical–quantum itself.

Analogously, one can see that a state ${\rho }_{{AB}}$ is classical–classical if, and only if, it is left invariant by at least one entanglement-breaking channel ${{\rm{\Lambda }}}_{{AB}}^{\mathrm{EB}}:= {{\rm{\Lambda }}}_{A}^{\mathrm{EB}}\otimes {{\rm{\Lambda }}}_{B}^{\mathrm{EB}}$ acting on both subsystems A and B [50], that is,

$$\begin{eqnarray}&&{\rho }_{{AB}}\in {C}_{{AB}}\quad \iff \quad \exists \text{an entanglement-breaking channel}\ \ {{\rm{\Lambda }}}_{{AB}}^{\mathrm{EB}}\ {\rm{such}}\,{\rm{that}}\ \ {{\rm{\Lambda }}}_{{AB}}^{\mathrm{EB}}[{\rho }_{{AB}}]={\rho }_{{AB}}.\end{eqnarray} \tag{ 41 }$$

Notice that, apart from the special case of local measurements, entanglement-breaking channels do not generally destroy the QCs content in a state ${\rho }_{{AB}}$, as they only vanquish its entanglement. Nevertheless, similarly to the case of local unitaries in section 2.3.3, it is the global change (or lack thereof) of the state ${\rho }_{{AB}}$ under the action of any entanglement-breaking channel that discerns the nature of the correlations in ${\rho }_{{AB}}$. Unlike the case of local unitaries, though, the invariance under suitable entanglement-breaking channels allows us to characterise both sets of classical–quantum and classical–classical states. The two statements in equations (40) and (41) thus formalise the defining property 6 of classical states, as illustrated in figure 3(e).

Definition 1 gives a negative characterisation of quantumly correlated states, i.e. quantumly correlated states are all the states that are not classical. The most natural quantitative extension of this is to consider the distance to the set of classical states as a measure of QCs. Given a distance ${D}_{\delta }$, we can then define geometric measures of one-sided and two-sided QCs of a bipartite state ${\rho }_{{AB}}$ as follows,

$$\begin{eqnarray}{Q}_{A}^{{G}_{\delta }}({\rho }_{{AB}}) & := & \mathop{\inf }\limits_{{\chi }_{{AB}}\in {C}_{A}}{D}_{\delta }({\rho }_{{AB}},{\chi }_{{AB}}),\\ {Q}_{{AB}}^{{G}_{\delta }}({\rho }_{{AB}}) & := & \mathop{\inf }\limits_{{\chi }_{{AB}}\in {C}_{{AB}}}{D}_{\delta }({\rho }_{{AB}},{\chi }_{{AB}}),\end{eqnarray} \tag{ 45 }$$

where δ labels the particular distance functional ${D}_{\delta }$ on the set of states ${D}_{{AB}}$, and the superscript ${G}_{\delta }$ indicates that the measures are of the geometric type with respect to this chosen distance.

By construction, the geometric measures are zero only for classical states, hence requirement (i) is satisfied. Requirements (ii), (iv), and (v) are also satisfied for any choice of ${D}_{\delta }$ that is contractive under the action of CPTP operations, i.e. such that

$$\begin{eqnarray}&&{D}_{\delta }\left((,{\rm{\Lambda }}[\rho ],{\rm{\Lambda }}[\sigma ],)\right)\leqslant {D}_{\delta }(\rho ,\sigma ),\end{eqnarray} \tag{ 46 }$$

for any two states ρ and σ and any CPTP operation Λ. Although contractivity may not be necessary to satisfy these properties, there is good reason to restrict to contractive distances. Distances are used to quantify the distinguishability between quantum states, and non-invertible CPTP operations are the representation of noise [77]. Intuitively, one expects noise to never increase the distinguishability of quantum states, hence the distances that contract through any CPTP operation are the mathematical realisation of this. We also note that, in a resource theoretic context, any contractive distance will induce a geometric measure of QCs that is monotonically non-increasing under the action of the free operations, regardless of their exact specification, purely by virtue of the fact that the free operations cannot create QCs. Table 2 summarises a selection of widely adopted contractive distances in quantum information theory, whose corresponding geometric QCs measures will be discussed in section 3.3.

Table 2.  Some distances ${D}_{\delta }$ between two quantum states ρ and σ. Details on the definitions and discussions of contractivity can be found in [77–79].

Distance	${D}_{\delta }$	Formula	Contractive
Relative entropy	${D}_{\mathrm{RE}}$	${ \mathcal S }(\rho \parallel \sigma )={\rm{Tr}}\left((,\rho \mathrm{log}\rho -\rho \mathrm{log}\sigma ,)\right)$	Yes
($p\mathrm{th}$ power) Schatten p-norm	Dp	$\parallel \rho -\sigma {\parallel }_{p}^{p}={\rm{Tr}}\left((,| \rho -\sigma {| }^{p},)\right)$	Only for p = 1
Infidelity	${D}_{F}$	$1-{\left[[,{\rm{Tr}}\left((,\sqrt{\sqrt{\sigma }\rho \sqrt{\sigma }},)\right),]\right]}^{2}$	Yes
Squared Bures	${D}_{\mathrm{Bu}}$	$2\left[[,1-{\rm{Tr}}\left((,\sqrt{\sqrt{\sigma }\rho \sqrt{\sigma }},)\right),]\right]$	Yes
Squared Hellinger	${D}_{\mathrm{He}}$	$2\left[[,1-{\rm{Tr}}\left((,\sqrt{\rho }\sqrt{\sigma },)\right),]\right]$	Yes

Finally, there is a subtlety regarding requirement (iii) for the geometric measures of QCs. To each geometric measure of QCs one can associate a corresponding measure of entanglement, defined in terms of the distance to the separable states. Specifically, for a bipartite state ${\rho }_{{AB}}$, the geometric measure of entanglement given a particular choice of distance ${D}_{\delta }$ is [16]

$$\begin{eqnarray}&&{E}^{{G}_{\delta }}({\rho }_{{AB}}):= \mathop{\inf }\limits_{{\sigma }_{{AB}}\in {S}_{{AB}}}{D}_{\delta }({\rho }_{{AB}},{\sigma }_{{AB}}),\end{eqnarray} \tag{ 47 }$$

where ${S}_{{AB}}$ is the set of separable states defined in equation (3). Because of the hierarchy of equation (12), as depicted in figure 2(a), it generally holds that

$$\begin{eqnarray}&&{Q}_{{AB}}^{{G}_{\delta }}({\rho }_{{AB}})\geqslant {Q}_{A}^{{G}_{\delta }}({\rho }_{{AB}})\geqslant {E}^{{G}_{\delta }}({\rho }_{{AB}}).\end{eqnarray} \tag{ 48 }$$

For pure states ${\left|| ,\psi ,\rangle \right.\rangle }_{{AB}}$, one would naturally expect that

$$\begin{eqnarray}&&{Q}_{{AB}}^{{G}_{\delta }}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})={Q}_{A}^{{G}_{\delta }}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})={E}^{{G}_{\delta }}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}}),\end{eqnarray} \tag{ 49 }$$

because the set of pure classical and pure separable states all reduce to the set of pure product states. However, since the infima in equations (45) and (47) are, respectively, over all (not necessarily pure) classical and separable states, equation (49) does not hold trivially. Instead, one must show requirement (iii) on a case-by-case basis for each choice of distance ${D}_{\delta }$.

Another intuitive aspect of QCs, outlined by definition 2, is that a quantumly correlated state is perturbed by any possible complete rank one LPM. If we consider a complete rank one LPM on subsystem A, we know that ${\pi }_{A}[{\rho }_{{AB}}]$ as defined in equation (18) is necessarily different to ${\rho }_{{AB}}$ if QCs are present. It is then meaningful to think that, provided one identifies the least perturbing LPM ${\pi }_{A}$, the more distinguished ${\pi }_{A}[{\rho }_{{AB}}]$ is from ${\rho }_{{AB}}$, the more QCs were present originally in ${\rho }_{{AB}}$. By quantifying this distinguishability as the distance between ${\rho }_{{AB}}$ and the closest (i.e., least perturbed) output state ${\pi }_{A}[{\rho }_{{AB}}]$, we can define the measurement induced geometric measures of QCs of a state ${\rho }_{{AB}}$. Given a distance ${D}_{\delta }$, these are defined as

$$\begin{eqnarray}{Q}_{A}^{{M}_{\delta }}({\rho }_{{AB}}) & := & \mathop{\inf }\limits_{{\pi }_{A}}{D}_{\delta }\left((,{\rho }_{{AB}},{\pi }_{A}[{\rho }_{{AB}}],)\right),\\ {Q}_{{AB}}^{{M}_{\delta }}({\rho }_{{AB}}) & := & \mathop{\inf }\limits_{{\pi }_{{AB}}}{D}_{\delta }\left((,{\rho }_{{AB}},{\pi }_{{AB}}[{\rho }_{{AB}}],)\right),\end{eqnarray} \tag{ 50 }$$

where ${M}_{\delta }$ denotes the measurement induced geometric type for a corresponding distance ${D}_{\delta }$, and we have considered LPMs on both subsystems A and B as defined in equation (19) in the case of two-sided measures.

Once more, we can choose any distance on the set of quantum states to define a measure within this family, with some common examples listed in table 2. The minimisation must be performed only over complete rank one LPMs, which represent the most fine grained projective measurements since each projector corresponds to a particular outcome and no outcomes are merged into a higher rank projector (more trivially, without restricting to rank one projectors one would always satisfy the infima in equation (50) by choosing as projectors, respectively, ${{\mathbb{I}}}_{A}$ and ${{\mathbb{I}}}_{A}\otimes {{\mathbb{I}}}_{B}$). We can use the fact that ${\pi }_{A}[{\rho }_{{AB}}]\in {C}_{A}$ and ${\pi }_{{AB}}[{\rho }_{{AB}}]\in {C}_{{AB}}$, as we have already discussed in equations (20) and (21), to see that

$$\begin{eqnarray}&&{Q}_{A}^{{G}_{\delta }}({\rho }_{{AB}})\leqslant {Q}_{A}^{{M}_{\delta }}({\rho }_{{AB}}),\quad {\rm{and}}\quad {Q}_{{AB}}^{{G}_{\delta }}({\rho }_{{AB}})\leqslant {Q}_{{AB}}^{{M}_{\delta }}({\rho }_{{AB}}).\end{eqnarray} \tag{ 51 }$$

For certain choices of ${D}_{\delta }$, these inequalities are in fact saturated, and we will highlight when this occurs in the following (section 3.3 and 3.4). It is immediate to see that requirement (i) for measurement induced geometric measures is satisfied because a state can be left invariant under a complete rank one LPM if, and only if, it is classical. Requirements (ii) and (iv) also hold for any contractive distance, however it is unknown whether this is true for requirement (v) as well. Again, requirement (iii) must be checked for each particular distance ${D}_{\delta }$.

An alternative viewpoint from which we can look at QCs is based on how much information about the state of a composite system can be extracted by accessing it locally. When only classical correlations are present, we can in principle extract all such information after a local measurement, while this is no longer the case when also QCs come into play. As discussed in section 2.3.1, LGMs ${\tilde{{\rm{\Pi }}}}_{A}$ and ${\tilde{{\rm{\Pi }}}}_{{AB}}$ represent recording the measurement information about ${\rho }_{{AB}}$ onto classical registers. It is intuitive, therefore, to adopt an information theoretic perspective and regard the minimal difference between the original information in the state ${\rho }_{{AB}}$ and the post-measurement information stored in the classical register as a measure of quantumness. Given an informational quantifier ${I}_{\epsilon }$, we can then define measurement induced informational (or simply informational) measures of QCs of a state ${\rho }_{{AB}}$ as

$$\begin{eqnarray}{Q}_{A}^{{\tilde{I}}_{\epsilon }}({\rho }_{{AB}}) & := & \mathop{\inf }\limits_{{\tilde{{\rm{\Pi }}}}_{A}}\left[[,{I}_{\epsilon }\left((,{\rho }_{{AB}},)\right)-{I}_{\epsilon }\left((,{\tilde{{\rm{\Pi }}}}_{A}[{\rho }_{{AB}}],)\right),]\right],\\ {Q}_{{AB}}^{{\tilde{I}}_{\epsilon }}({\rho }_{{AB}}) & := & \mathop{\inf }\limits_{{\tilde{{\rm{\Pi }}}}_{{AB}}}\left[[,{I}_{\epsilon }\left((,{\rho }_{{AB}},)\right)-{I}_{\epsilon }\left((,{\tilde{{\rm{\Pi }}}}_{{AB}}[{\rho }_{{AB}}],)\right),]\right].\end{eqnarray} \tag{ 52 }$$

A list of the most prominent choices of ${I}_{\epsilon }$ is presented in table 3. We elaborate more on these choices in section 3.5, also highlighting the relationships between informational and other types of measures of QCs.

Table 3.  Some informational quantifiers ${I}_{\epsilon }$ of a bipartite state ${\rho }_{{AB}}$ with reduced states ${\rho }_{A}={{\rm{Tr}}}_{B}({\rho }_{{AB}})$ and ${\rho }_{B}={{\rm{Tr}}}_{A}({\rho }_{{AB}})$.

Informational quantity	${I}_{\epsilon }$	Formula
(negative) von Neumann entropy	${I}_{{ \mathcal S }}$	$-{ \mathcal S }({\rho }_{{AB}})={\rm{Tr}}\left((,{\rho }_{{AB}}\mathrm{log}{\rho }_{{AB}},)\right)$
Mutual information	${I}_{{ \mathcal I }}$	${ \mathcal I }({\rho }_{{AB}})={ \mathcal S }({\rho }_{A})+{ \mathcal S }({\rho }_{B})-{ \mathcal S }({\rho }_{{AB}})$

The measures in equation (52) obey requirement (i) since for all and only the classical states there is always an LPM (which is within the class of all LGMs) that leaves them unchanged. However, one should be careful when choosing ${I}_{\epsilon }$ to ensure that ${Q}_{A}^{{\tilde{I}}_{\epsilon }}({\rho }_{{AB}})$ and ${Q}_{{AB}}^{{\tilde{I}}_{\epsilon }}({\rho }_{{AB}})$ are non-negative (hence the choice of the negative von Neumann entropy in table 3). Requirement (ii) holds for any ${I}_{\epsilon }$ invariant under local unitaries, which in turn should always hold as a measure of information or correlations is not expected to be dependent upon the local bases chosen. However, requirements (iii), (iv) and (v) must be checked on a case-by-case basis for each selection of ${I}_{\epsilon }$. Notice also that for some choices of I one may need to impose further restrictions to the allowed LGMs in the optimisations appearing in equation (52) in order to avoid trivial results, as we discuss in section 3.5.1.

We can explicitly consider the special case of informational measures with optimisation restricted over complete rank one LPMs, hence defining

$$\begin{eqnarray}{Q}_{A}^{{I}_{\epsilon }}({\rho }_{{AB}}) & := & \mathop{\inf }\limits_{{{\rm{\Pi }}}_{A}}\left[[,{I}_{\epsilon }\left((,{\rho }_{{AB}},)\right)-{I}_{\epsilon }\left((,{{\rm{\Pi }}}_{A}[{\rho }_{{AB}}],)\right),]\right],\\ {Q}_{{AB}}^{{I}_{\epsilon }}({\rho }_{{AB}}) & := & \mathop{\inf }\limits_{{{\rm{\Pi }}}_{{AB}}}\left[[,{I}_{\epsilon }\left((,{\rho }_{{AB}},)\right)-{I}_{\epsilon }\left((,{{\rm{\Pi }}}_{{AB}}[{\rho }_{{AB}}],)\right),]\right].\end{eqnarray} \tag{ 53 }$$

It then holds by construction that

$$\begin{eqnarray}&&{Q}_{A}^{{\tilde{I}}_{\epsilon }}({\rho }_{{AB}})\leqslant {Q}_{A}^{{I}_{\epsilon }}({\rho }_{{AB}}),\quad {\rm{and}}\quad {Q}_{{AB}}^{{\tilde{I}}_{\epsilon }}({\rho }_{{AB}})\leqslant {Q}_{{AB}}^{{I}_{\epsilon }}({\rho }_{{AB}}).\end{eqnarray} \tag{ 54 }$$

To be more precise, invoking Naimark's theorem as in section 2.3.1, we recall that an LGM on ${\rho }_{{AB}}$ can equivalently be performed by embedding ${\rho }_{{AB}}$ into a larger space and carrying out an LPM. This means that

$$\begin{eqnarray}&&{Q}_{A}^{{\tilde{I}}_{\epsilon }}({\rho }_{{AB}})=\mathop{\inf }\limits_{{A}^{\prime }}{Q}_{A^{\prime} }^{{I}_{\epsilon }}({\rho }_{A^{\prime} B}),\quad {\rm{and}}\quad {Q}_{{AB}}^{{\tilde{I}}_{\epsilon }}({\rho }_{{AB}})=\mathop{\inf }\limits_{A^{\prime} ,B^{\prime} }{Q}_{A^{\prime} B^{\prime} }^{{I}_{\epsilon }}({\rho }_{A^{\prime} B^{\prime} }),\end{eqnarray} \tag{ 55 }$$

or in words that the one-sided LGM based informational measures are obtained as a minimisation of the corresponding one-sided LPM based informational measures over all embeddings of A into a larger system $A^{\prime}$, and similarly for the two-sided measures considering embeddings of both A and B in larger systems $A^{\prime}$ and $B^{\prime}$, respectively.

Taking care again to ensure non-negativity, the LPM subclass of informational measures obey requirement (i) and requirement (ii) if ${I}_{\epsilon }$ is invariant under local unitaries, but all the other requirements must be checked for each ${I}_{\epsilon }$ on a case-by-case basis. The meaning of the complete rank one LPM based informational measures in equations (53) is even more intuitive. Indeed, by recalling that the output of the quantum-to-classical maps ${{\rm{\Pi }}}_{A}$ (resp., ${{\rm{\Pi }}}_{{AB}}$) can be considered equivalent to the post-measurement states ${\pi }_{A}$ (resp., ${\pi }_{{AB}}$), we have that such informational measures capture directly the minimum amount of change in the information content of the state ${\rho }_{{AB}}$ itself (without the need for invoking a classical register) before and after a local von Neumann measurement. This reproduces the original spirit of the quantum discord [6], which can be classified indeed as an informational measure, as we show in section 3.5.

Definition 3 suggests that measures of QCs can be readily defined from measures of entanglement. In particular, the QCs content of a state ${\rho }_{{AB}}$ can be quantified in terms of the minimum entanglement created between the system AB and an ancillary system, given by $A^{\prime}$ for the one-sided case, or by the composite $A^{\prime} B^{\prime}$ for the two-sided case, in the pre-measurement state corresponding to a least perturbing complete rank one LPM on A or on both A and B, respectively. Given a specific bipartite entanglement monotone ${E}^{\zeta }$, this conceptual framework initially put forward in [35, 36] defines the entanglement activation measures of QCs of a state ${\rho }_{{AB}}$,

$$\begin{eqnarray}{Q}_{A}^{{E}^{\zeta }}({\rho }_{{AB}}) & := & \mathop{\inf }\limits_{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}{E}_{{AB}:A^{\prime} }^{\zeta }\left((,\rho {^{\prime} }_{{ABA}^{\prime} }^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}},)\right),\\ {Q}_{{AB}}^{{E}^{\zeta }}({\rho }_{{AB}}) & := & \mathop{\inf }\limits_{\{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}}{E}_{{AB}:A^{\prime} B^{\prime} }^{\zeta }\left((,\rho {^{\prime} }_{{ABA}^{\prime} B^{\prime} }^{\{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}},)\right),\end{eqnarray} \tag{ 56 }$$

where the optimisation is over the local bases of the pre-measurement interaction as detailed in section 2.3.2, and the pre-measurement states $\rho {^{\prime} }_{{ABA}^{\prime} }^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}$ and $\rho {^{\prime} }_{{ABA}^{\prime} B^{\prime} }^{\{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}}$ are specifically defined in equations (22) and (25); notice that we indicate the explicit bipartition $X\,:Y$ with respect to which entanglement is being calculated with the notation ${E}_{X:Y}^{\zeta }({\rho }_{{XY}})$.

One of the key strengths of this approach is that there is a huge variety of entanglement measures ${E}^{\zeta }$ available in the quantum information literature [16] (we list a small selection of choices in table 4): for each of those, equation (56) define corresponding measures of more general one-sided and two-sided QCs. Crucially, it turns out that this correspondence respects the hierarchy of non-classical correlations illustrated in figures 1 and 2(a). Indeed, it was proven in [28] that

$$\begin{eqnarray}&&{Q}_{{AB}}^{{E}^{\zeta }}({\rho }_{{AB}})\geqslant {Q}_{A}^{{E}^{\zeta }}({\rho }_{{AB}})\geqslant {E}_{A:B}^{\zeta }({\rho }_{{AB}}),\end{eqnarray} \tag{ 57 }$$

for all bipartite states ${\rho }_{{AB}}$ and any ${E}^{\zeta }$ monotonically non-increasing under LOCC. This follows because the original state ${\rho }_{{AB}}$ can be obtained from the pre-measurement states in equation (56) by means of an LOCC with respect to the system versus ancilla(e) split [28]. The inequalities in equation (57) all turn into equalities for pure states ${\left|| ,\psi ,\rangle \right.\rangle }_{{AB}}$, which means that requirement (iii) is satisfied for all entanglement activation measures of QCs [28]. It is rightful to interpret equation (57) as a quantitative indication that QCs truly go beyond entanglement, in a very precise sense, and reduce to it in the case of pure states, as it should be. Requirement (i) holds by construction due to the characterisation of classical states according to property 3, and requirements (ii) and (iv) are satisfied as well for any entanglement activation measure of QCs defined from a valid entanglement monotone ${E}^{\zeta }$. Requirement (v) has not been investigated in general and remains to be checked on a case-by-case basis.

Table 4.  Some entanglement monotones ${E}^{\zeta }$ for a bipartite state ${\rho }_{{AB}}$.

Entanglement measure	${E}^{\zeta }$	Formula
Geometric measure(s)	${E}^{{G}_{\delta }}$	${E}^{{G}_{\delta }}({\rho }_{{AB}})=\mathop{\inf }\limits_{{\sigma }_{{AB}}\in {S}_{{AB}}}{D}_{\delta }({\rho }_{{AB}},{\sigma }_{{AB}})$
Distillable entanglement	Ed	See [3, 16]
Entanglement negativity	EN	${E}^{N}({\rho }_{{AB}})=\parallel {\rho }_{{AB}}^{{T}_{A}}{\parallel }_{1}-1$
Entanglement of formation	Ef	${E}^{f}({\rho }_{{AB}})=\mathop{\inf }\limits_{\genfrac{}{}{0em}{}{\{{p}_{i},{\left|| ,{\psi }_{i},\rangle \right.\rangle }_{{AB}}\},}{{\rho }_{{AB}}={\sum }_{i}{p}_{i}\left|| ,{\psi }_{i},\rangle \right.\rangle {\left.\langle \langle ,{\psi }_{i},| \right|}_{{AB}}}}{\sum }_{i}{p}_{i}\ { \mathcal S }\left((,{{\rm{Tr}}}_{B}(\left|| ,{\psi }_{i},\rangle \right.\rangle \,{\left.\langle \langle ,{\psi }_{i},| \right|}_{{AB}}),)\right)$

We discuss some specific examples of entanglement activation type measures of QCs in section 3.6.

As outlined by definition 4, QCs give also rise to the unavoidable disturbance of a bipartite quantum state in response to the action of non-degenerate unitary operations on one of the subsystems. This qualitative aspect of QCs can be turned into a QCs quantifier by considering how far a given bipartite state ${\rho }_{{AB}}$ is from being invariant under a non-degenerate local unitary operation, i.e., by considering the distinguishability between ${\rho }_{{AB}}$ and its image after a least disturbing non-degenerate local unitary. For a given distance D_δ, the (one-sided) unitary response measure of QCs of a state ${\rho }_{{AB}}$ is then defined as [39–41, 80]

$$\begin{eqnarray}&&{Q}_{A}^{{U}_{\delta }^{{\rm{\Gamma }}}}({\rho }_{{AB}}):= \mathop{\inf }\limits_{{U}_{A}^{{\rm{\Gamma }}}}{D}_{\delta }\left((,{\rho }_{{AB}},{U}_{A}^{{\rm{\Gamma }}}[{\rho }_{{AB}}],)\right),\end{eqnarray} \tag{ 58 }$$

where the optimisation is over all local unitaries ${U}_{A}^{{\rm{\Gamma }}}$ with non-degenerate spectrum Γ, as defined in equation (28).

Among all possible choices of the spectrum Γ, a special role is played by the harmonic spectrum ${{\rm{\Gamma }}}^{\star }$ given by the d_Ath complex roots of unity, with d_A being the dimension of subsystem A. The corresponding local unitary operations, also known as root-of-unity operations, will be denoted simply by ${U}_{A}\equiv {U}_{A}^{{{\rm{\Gamma }}}^{\star }}$ (dropping the spectrum superscript). They may be expected to be the most informative ones that we can restrict to in order to quantify QCs, in the sense that the more the unitaries in the aforementioned optimisation are fully non-degenerate (with no spacing between any two eigenvalues becoming arbitrarily small) the better the corresponding response measures should be able to discern between different states [39–41, 81]. For this reason, we will primarily focus on the special case of unitary response measures with optimisation restricted to harmonic spectra,

$$\begin{eqnarray}&&{Q}_{A}^{{U}_{\delta }}({\rho }_{{AB}}):= \mathop{\inf }\limits_{{U}_{A}}{D}_{\delta }\left((,{\rho }_{{AB}},{U}_{A}[{\rho }_{{AB}}],)\right).\end{eqnarray} \tag{ 59 }$$

The unitary response measures of QCs clearly satisfy requirement (i) since any classical–quantum state is invariant under a non-degenerate local unitary [39–41]. Moreover, they satisfy requirement (ii) and, provided D_δ is contractive, requirement (iv) [41], whereas it is still an open question if they comply with requirement (v). For pure states ${\left|| ,\psi ,\rangle \right.\rangle }_{{AB}}$, any measure ${Q}_{A}^{{U}_{\delta }}$ reduces to the corresponding 'entanglement of response' [81],

$$\begin{eqnarray}&&{E}^{{U}_{\delta }}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}}):= \mathop{\inf }\limits_{{U}_{A}}{D}_{\delta }\left((,\left|| ,\psi ,\rangle \right.\rangle {\left.\langle \langle ,\psi ,| \right|}_{{AB}},{U}_{A}[\left|| ,\psi ,\rangle \right.\rangle {\left.\langle \langle ,\psi ,| \right|}_{{AB}}],)\right),\end{eqnarray} \tag{ 60 }$$

which entails that requirement (iii) is satisfied as well for any unitary response measure of QCs.

Examples of measures ${Q}_{A}^{{U}_{\delta }}$ for typical distances D_δ as given in table 2, and their connections to geometric and measurement induced geometric measures will be explored in section 3.7. We finally remark that, by virtue of the discussion in section 2.3.3, the unitary response approach does not appear to be straightforwardly extendible to define two-sided measures of QCs, as the latter would fail requirement (i), unlike the case of all the other types of measures covered in this review.

Following definition 5, QCs can be quantified in terms of measures of quantum coherence, minimised over all local bases. Measures of quantum coherence have been studied quite extensively in the last few years, see e.g. [42–49, 69–71, 82, 83] and the very recent review [84]. While several connections between coherence, entanglement, and QCs have been pointed out quite recently, including quantitative interconversion schemes [46, 47, 49, 85, 86] somehow analogous to the entanglement activation framework [35, 36], a consistent definition of QCs quantifiers in terms of coherence has not been reported in full generality (to our knowledge), and will be introduced here.

For a single system, consider a measure of coherence ${C}^{\eta }$ with respect to a fixed reference basis $\{\left|| ,i,\rangle \right.\rangle \}$, that is, a real, non-negative function ${C}^{\eta }(\rho )$ which vanishes on, and only on, the corresponding set ${I}^{\{\left|| ,i,\rangle \right.\rangle \}}$ of incoherent states (see section 2.3.4), and is monotonically non-increasing under a suitable set of incoherent operations. The incoherent operations may be, for instance, those defined in [43], which admit an operator sum representation where all Kraus operators map the set of incoherent states into itself, or a more restricted set such as the so-called covariant operations, which have been defined in the context of the resource theory of asymmetry [69]; we defer the reader to recent studies where merits and setbacks of various proposals for incoherent operations are discussed [44, 45, 70–73]. We list a sample of coherence measures ${C}^{\eta }$ of relevance in various resource theoretic contexts in table 5.

Table 5.  Some coherence measures for a state ρ with respect to a reference basis $\{\left|| ,i,\rangle \right.\rangle \}$. The first four [42, 43, 45, 49] are monotones under the more general incoherent operations of [43], while the last two [70, 82, 83], in which the reference basis is identified as the eigenbasis of the observable $K={\sum }_{i}{k}_{i}\left|| ,i,\rangle \right.\rangle \,\left.\langle \langle ,i,| \right|$ with non-degenerate spectrum ${\rm{\Gamma }}=\{{k}_{i}\}$, are monotones under the more restricted set of phase covariant operations defined in asymmetry theory [69]. Furthermore, in the first row we denote by $\pi [\rho ]$ the full dephasing in the reference basis, $\pi [\rho ]={\sum }_{i}\left|| ,i,\rangle \right.\rangle \,\left.\langle \langle ,i,| \right|\rho \left|| ,i,\rangle \right.\rangle \,\left.\langle \langle ,i,| \right|$, in the third row we evaluate the ℓ₁ norm for ρ written in the reference basis, in the fourth row we indicate by τ an arbitrary state, while in the final row we write ρ in its spectral decomposition, $\rho ={\sum }_{n}{q}_{n}\left|| ,{\psi }_{n},\rangle \right.\rangle \,\left.\langle \langle ,{\psi }_{n},| \right|$.

Coherence measure	${C}^{\eta }$	Formula
Relative entropy	${C}^{\mathrm{RE}}$	${C}^{\mathrm{RE}}(\rho )=\mathop{\inf }\limits_{\chi \in {I}^{\{\left|| ,i,\rangle \right.\rangle \}}}{D}_{\mathrm{RE}}(\rho ,\chi )={ \mathcal S }\left((,\pi [\rho ],)\right)-{ \mathcal S }(\rho )$
Infidelity	CF	${C}^{F}(\rho )=1-\mathop{\sup }\limits_{\chi \in {I}^{\{\left|| ,i,\rangle \right.\rangle \}}}F(\rho ,\chi )$
ℓ1 norm	${C}^{{{\ell }}_{1}}$	${C}^{{{\ell }}_{1}}(\rho )=\parallel \rho {\parallel }_{{{\ell }}_{1}}^{\{\left|| ,i,\rangle \right.\rangle \}}-1={\sum }_{i\ne j}\,| {\rho }_{{ij}}|$
Robustness	${C}^{\mathrm{Ro}}$	${C}^{\mathrm{Ro}}(\rho )=\mathop{\min }\limits_{\tau }\left\{\{,s\geqslant 0\ \ ,| \right|\ \ \tfrac{\rho +s\tau }{1+s}\in {I}^{\{\left|| ,i,\rangle \right.\rangle \}}\}$
Wigner–Yanase skew information	${C}^{\mathrm{WY}}$	${C}^{\mathrm{WY}}(\rho ,K)=-\tfrac{1}{2}{\rm{Tr}}\left((,{\left[[,\sqrt{\rho },K,]\right]}^{2},)\right)$
Quantum Fisher information	${C}^{\mathrm{QF}}$	${C}^{\mathrm{QF}}(\rho ,K)=4{\sum }_{m\lt n,{q}_{m}+{q}_{n}\ne 0}\tfrac{{\left((,{q}_{m}-{q}_{n},)\right)}^{2}}{{q}_{m}+{q}_{n}}{\left|| ,\left.\langle \langle ,{\psi }_{m},| \right|K\left|| ,{\psi }_{n},\rangle \right.\rangle ,| \right|}^{2}$

Given now a bipartite system AB in the state ${\rho }_{{AB}}$, and fixing local reference bases $\{{\left|| ,a,\rangle \right.\rangle }_{A}\}$ for subsystem A and $\{{\left|| ,b,\rangle \right.\rangle }_{B}\}$ for subsystem B as in section 2.3.4, assume that to a chosen coherence measure ${C}^{\eta }$ we can formally associate: a one-sided local coherence quantifier ${C}_{A}^{\eta \ \{{\left|| ,a,\rangle \right.\rangle }_{A}\}}({\rho }_{{AB}})$ with the requirements of vanishing on (and only on) the set ${I}_{A}^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}$ of incoherent-quantum states defined in equation (34), and of being monotonic under (the chosen set of) local incoherent operations on A and arbitrary local operations on B; and a two-sided local coherence quantifier ${C}_{{AB}}^{\eta \ \{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}}({\rho }_{{AB}})$ with the requirements of vanishing on (and only on) the set ${I}_{{AB}}^{\{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}}$ of incoherent-incoherent states defined in equation (35), and of being monotonic under (the chosen set of) local incoherent operations on A and B. In particular, the two-sided quantifier can be defined as the standard extension of ${C}^{\eta }$ to bipartite systems, choosing the product basis $\{{\left|| ,a,\rangle \right.\rangle }_{A}\otimes {\left|| ,b,\rangle \right.\rangle }_{B}\}$ as reference basis for the composite AB [44, 48, 49]. Notice however that, depending on the definition of ${C}^{\eta }$, it may not be straightforwardly possible to construct both one-sided and two-sided corresponding quantifiers of local coherence; we will discuss this in more detail in section 3.8.

For a given coherence quantifier ${C}^{\eta }$, we then define the coherence based measures of QCs of a state ${\rho }_{{AB}}$ as

$$\begin{eqnarray}{Q}_{A}^{{C}^{\eta }}({\rho }_{{AB}}) & := & \mathop{\inf }\limits_{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}{C}_{A}^{\eta \ \{{\left|| ,a,\rangle \right.\rangle }_{A}\}}({\rho }_{{AB}}),\\ {Q}_{{AB}}^{{C}^{\eta }}({\rho }_{{AB}}) & := & \mathop{\inf }\limits_{\{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}}{C}_{{AB}}^{\eta \ \{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}}({\rho }_{{AB}}),\end{eqnarray} \tag{ 61 }$$

where the minimisation is over the local reference bases $\{{\left|| ,a,\rangle \right.\rangle }_{A}\}$ of A and $\{{\left|| ,b,\rangle \right.\rangle }_{B}\}$ of B.

By construction, due to property 5 and the constraints imposed on the local coherence quantifiers above, coherence based measures of QCs satisfy requirements (i) and (iv), and also requirement (ii), which holds manifestly in the two-sided case due to the involved minimisation over local bases on both parties and is implied by (iv) in the one-sided case. However, there are no general results yet concerning requirements (iii) and (v), which have to be checked on a case-by-case basis.

In section 3.8 we show how the coherence based approach encompasses some relevant quantifiers of QCs, and discuss their interplay with other types of measures.

The last facet of QCs that we take into account is the following: when the QCs between two subsystems are small, almost all the information about the overall bipartite system is locally recoverable after performing a measurement on one of the subsystems. Since every entanglement-breaking channel can be written as a sequence of a measurement map followed by a preparation (recovery) map, as discussed in section 2.3.5, such a facet of QCs can be quantitatively captured by how far the state ${\rho }_{{AB}}$ of the overall bipartite system is from being a fixed point of an entanglement-breaking channel [50], along the lines of definition 6. Given a distance D_δ, we can then define the recoverability type measures of QCs as

$$\begin{eqnarray}{Q}_{A}^{{R}_{\delta }}({\rho }_{{AB}}) & := & \mathop{\inf }\limits_{{{\rm{\Lambda }}}_{A}^{\mathrm{EB}}}{D}_{\delta }\left((,{\rho }_{{AB}},{{\rm{\Lambda }}}_{A}^{\mathrm{EB}}[{\rho }_{{AB}}],)\right),\\ {Q}_{{AB}}^{{R}_{\delta }}({\rho }_{{AB}}) & := & \mathop{\inf }\limits_{{{\rm{\Lambda }}}_{{AB}}^{\mathrm{EB}}}{D}_{\delta }\left((,{\rho }_{{AB}},{{\rm{\Lambda }}}_{{AB}}^{\mathrm{EB}}[{\rho }_{{AB}}],)\right),\end{eqnarray} \tag{ 62 }$$

where the optimisation is over the convex set of entanglement-breaking channels ${{\rm{\Lambda }}}_{A}^{\mathrm{EB}}$ acting locally on A and ${{\rm{\Lambda }}}_{{AB}}^{\mathrm{EB}}$ acting locally on A and on B, respectively for one-sided and two-sided measures.

The recoverability based measures comply with requirement (i) as all and only the classical state are fixed points of some entanglement-breaking channel. They further satisfy requirements (ii) and (iv) provided D_δ is a contractive distance, while requirements (iii) and requirement (v) have to be tested for each D_δ. Observe that, given an arbitrary state ${\rho }_{{AB}}$, the recoverability based measures of equation (62) are in general smaller or equal than the corresponding measurement induced geometric measures of equation (50) defined via the same ${D}_{\delta }$, since every LPM is in particular a local entanglement-breaking channel.

Instances of recoverability measures of QCs as defined in recent literature [50] are presented in section 3.9.

The first definition of a geometric measure of QCs was given in [87, 88], where the relative entropy ${D}_{\mathrm{RE}}$ (see table 2) was considered. Although not strictly a distance [77], the relative entropy (or Kullback-Leibler divergence) is a widely used tool in classical information theory to measure the distinguishability between two probability distributions, and it is naturally extended to quantum states [89]. The relative entropy is contractive [90, 91], therefore we need only to check requirement (iii) for the corresponding QCs measures ${Q}_{A}^{{G}_{\mathrm{RE}}}$ and ${Q}_{{AB}}^{{G}_{\mathrm{RE}}}$. The latter is verified, as for pure states ${\left|| ,\psi ,\rangle \right.\rangle }_{{AB}}$ it holds

$$\begin{eqnarray}&&{Q}_{A}^{{G}_{\mathrm{RE}}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})={Q}_{{AB}}^{{G}_{\mathrm{RE}}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})={E}^{{G}_{\mathrm{RE}}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})={ \mathcal S }({\rho }_{A})={ \mathcal S }({\rho }_{B}),\end{eqnarray} \tag{ 63 }$$

where ${ \mathcal S }({\rho }_{A})={ \mathcal S }({\rho }_{B})$ amounts to the canonical measure of entanglement for the pure state ${\left|| ,\psi ,\rangle \right.\rangle }_{{AB}}$, known as the entanglement entropy [77, 92], with the marginal state ${\rho }_{A}$ (${\rho }_{B}$) obtained from $\left|| ,\psi ,\rangle \right.\rangle {\left.\langle \langle ,\psi ,| \right|}_{{AB}}$ by partial trace over B (A).

In [93] it was shown that the closest classical state ${\chi }_{{AB}}$ to any state ${\rho }_{{AB}}$, which solves the optimisation in equation (45) for the relative entropy distance, is always the one obtained as the output of least perturbing complete rank one LPMs ${\pi }_{A}[{\rho }_{{AB}}]$ and ${\pi }_{{AB}}[{\rho }_{{AB}}]$ for the one-sided and two-sided measures, respectively. This implies that the geometric and measurement induced geometric measures based upon the relative entropy distance coincide, i.e. 

$$\begin{eqnarray}&&{Q}_{A}^{{G}_{\mathrm{RE}}}({\rho }_{{AB}})={Q}_{A}^{{M}_{\mathrm{RE}}}({\rho }_{{AB}}),\quad {\rm{and}}\quad {Q}_{{AB}}^{{G}_{\mathrm{RE}}}({\rho }_{{AB}})={Q}_{{AB}}^{{M}_{\mathrm{RE}}}({\rho }_{{AB}}).\end{eqnarray} \tag{ 64 }$$

Furthermore, there is an agreement with the LPM informational measures based on the negative von Neumann entropy [87, 94] (see table 3)

$$\begin{eqnarray}{Q}_{A}^{{G}_{\mathrm{RE}}}({\rho }_{{AB}}) & = & \mathop{\inf }\limits_{{{\rm{\Pi }}}_{A}}\left[[,{ \mathcal S }({{\rm{\Pi }}}_{A}[{\rho }_{{AB}}])-{ \mathcal S }({\rho }_{{AB}}),]\right]={Q}_{A}^{{I}_{{ \mathcal S }}}({\rho }_{{AB}}),\\ {Q}_{{AB}}^{{G}_{\mathrm{RE}}}({\rho }_{{AB}}) & = & \mathop{\inf }\limits_{{{\rm{\Pi }}}_{{AB}}}\left[[,{ \mathcal S }({{\rm{\Pi }}}_{{AB}}[{\rho }_{{AB}}])-{ \mathcal S }({\rho }_{{AB}}),]\right]={Q}_{{AB}}^{{I}_{{ \mathcal S }}}({\rho }_{{AB}}),\end{eqnarray} \tag{ 65 }$$

as well as with the entanglement activation measures based upon relative entropy and distillable entanglement [35, 36]

$$\begin{eqnarray}&&{Q}_{A}^{{G}_{\mathrm{RE}}}({\rho }_{{AB}})={Q}_{A}^{{E}^{{G}_{\mathrm{RE}}}}({\rho }_{{AB}})={Q}_{A}^{{E}^{d}}({\rho }_{{AB}}),\quad {\rm{and}}\quad {Q}_{{AB}}^{{G}_{\mathrm{RE}}}({\rho }_{{AB}})={Q}_{{AB}}^{{E}^{{G}_{\mathrm{RE}}}}({\rho }_{{AB}})={Q}_{{AB}}^{{E}^{d}}({\rho }_{{AB}}),\end{eqnarray} \tag{ 66 }$$

and also with the coherence based measures in terms of relative entropy ${Q}_{A}^{{C}^{\mathrm{RE}}}({\rho }_{{AB}})$ and ${Q}_{{AB}}^{{C}^{\mathrm{RE}}}({\rho }_{{AB}})$, defined later in equation (128) [43, 47, 93, 95].

There is clearly a quite remarkable convergence on the one-sided and two-sided measures ${Q}_{A}^{{G}_{\mathrm{RE}}}$ and ${Q}_{{AB}}^{{G}_{\mathrm{RE}}}$, which are also known in the literature as 'relative entropy of discord' and 'relative entropy of quantumness' respectively [36, 87, 88, 93], across a range of different (generally inequivalent) categories of QCs with accordingly different physical interpretations.

As it happens customarily in quantum information theory, however, the best behaved quantities are often the most difficult to compute [96]. In the case of the geometric measures of QCs based on relative entropy, ${Q}_{A}^{{G}_{\mathrm{RE}}}$ and ${Q}_{{AB}}^{{G}_{\mathrm{RE}}}$, analytical formulae are available only for special classes of states, such as Bell diagonal states of two qubits [97] and highly symmetric (i.e., Werner and isotropic) states of two qudits [98]. For both of these classes of states, one-sided and two-sided measures of QCs coincide.

By using the 2-norm induced distance D₂ (see table 2), equal to the squared Hilbert–Schmidt distance, another geometric quantifier of one-sided QCs, corresponding to ${Q}_{A}^{{G}_{2}}({\rho }_{{AB}})$ as defined in equation (45), was proposed in [25], where a simple closed form of this quantity for arbitrary states of two qubits was also found. Extensions beyond two qubits were subsequently derived, including in particular analytical formulae for all states where subsystem A is a qubit while the dimension of subsystem B is arbitrary [99–102], and an operational interpretation was provided by linking ${Q}_{A}^{{G}_{2}}({\rho }_{{AB}})$ to the task of remote state preparation [103] (see section 4.1.5 for a critical discussion). It was later shown that [99]

$$\begin{eqnarray}&&{Q}_{A}^{{G}_{2}}({\rho }_{{AB}})={Q}_{A}^{{M}_{2}}({\rho }_{{AB}}),\end{eqnarray} \tag{ 67 }$$

while, if A is a qubit, it also holds that [39, 40]

$$\begin{eqnarray}&&{Q}_{A}^{{G}_{2}}({\rho }_{{AB}})=\displaystyle \frac{1}{4}{Q}_{A}^{{U}_{2}}({\rho }_{{AB}}).\end{eqnarray} \tag{ 68 }$$

Experimental quantifications of ${Q}_{A}^{{G}_{2}}({\rho }_{{AB}})$ have been carried out [103–106] and lower bounds provided [107–109]. The two-sided quantifier ${Q}_{{AB}}^{{G}_{2}}$ was defined in [110].

The Hilbert–Schmidt based geometric measure of QCs enjoyed considerable popularity due to its analytical simplicity and became initially known just as the 'geometric discord', but an important problem was subsequently identified, due to the fact that the Hilbert–Schmidt distance D₂ is not contractive in general, as mentioned in table 2. Although D₂ is still unitarily invariant, which implies requirement (ii), this means that requirements (iv) and (v) are not automatically satisfied for ${Q}_{A}^{{G}_{2}}$. Indeed, [111] provided a counter argument to show that ${Q}_{A}^{{G}_{2}}({\rho }_{{AB}})$ does not obey requirement (iv), which goes as follows. Consider appending an uncorrelated ancilla state ${\rho }_{C}$ to the bipartite state ${\rho }_{{AB}}$, it then holds that

$$\begin{eqnarray}&&{Q}_{A}^{{G}_{2}}({\rho }_{{AB}}\otimes {\rho }_{C})={Q}_{A}^{{G}_{2}}({\rho }_{{AB}}){\rm{Tr}}({\rho }_{C}^{2}),\end{eqnarray} \tag{ 69 }$$

due to the multiplicativity of the squared Hilbert–Schmidt norm on tensor products. Since ${\rm{Tr}}({\rho }_{C}^{2})$ is just the purity of ${\rho }_{C}$, we know that adding/removing an impure ancilla will decrease/increase ${Q}_{A}^{{G}_{2}}$. However, addition and removal of an ancilla is a reversible quantum operation on the unmeasured subsystem B, for which we require the one-sided QCs never to increase. Thus, ${Q}_{A}^{{G}_{2}}$ does not obey this crucial requirement. One proposed resolution to this particular problem was to rescale ${Q}_{A}^{{G}_{2}}({\rho }_{{AB}})$ by the purity of ${\rho }_{{AB}}$ [96], but in this case it is still possible to find other counterexamples showing that requirement (iv) is violated [112]. Another suggestion given in [113] was to exploit the link between the Hilbert–Schmidt geometric and measurement induced geometric measures in equation (67) and instead consider ${Q}_{A}^{{M}_{2}}(\sqrt{{\rho }_{{AB}}})$. While maintaining the analytical simplicity of ${Q}_{A}^{{G}_{2}}({\rho }_{{AB}})$, this remedied modification no longer experiences the problem in equation (69), and for pure states ${\left|| ,\psi ,\rangle \right.\rangle }_{{AB}}$ it reduces to ${Q}_{A}^{{M}_{2}}({\rho }_{{AB}})$ because $\sqrt{\left|| ,\psi ,\rangle \right.\rangle {\left.\langle \langle ,\psi ,| \right|}_{{AB}}}=\left|| ,\psi ,\rangle \right.\rangle {\left.\langle \langle ,\psi ,| \right|}_{{AB}}$. Finally, requirement (iii) is satisfied for ${Q}_{A}^{{G}_{2}}({\rho }_{{AB}})$, since for pure states one finds [100]

$$\begin{eqnarray}&&{Q}_{A}^{{G}_{2}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})=1-{\rm{Tr}}({\rho }_{A}^{2})=1-{\rm{Tr}}({\rho }_{B}^{2}),\end{eqnarray} \tag{ 70 }$$

which is half of the entanglement monotone known as the I-tangle [114]. The latter reduces to the squared entanglement negativity [115, 116] if A is a qubit, where we define the entanglement negativity ${E}^{N}({\rho }_{{AB}})$ as in table 4, with the suffix T_A denoting partial transposition with respect to subsystem A. More generally, $2{Q}_{A}^{{G}_{2}}({\rho }_{{AB}})\geqslant {[{E}^{N}({\rho }_{{AB}})]}^{2}$ for all two-qubit states ${\rho }_{{AB}}$ [117], but this is not true anymore for higher dimensional states [118]. Geometric measures of QCs based on the Hilbert–Schmidt distance have been studied for two-mode Gaussian states in [96, 119] (specifically, in the measurement induced geometric approach); however, the only classical Gaussian states are product states [120, 121], which makes any geometric approach to the quantification of their QCs ambiguous (i.e., geometric quantum and total correlations collapse to the same quantity when restricting distances to the subsets ${C}_{A}$ or ${C}_{{AB}}$ and ${P}_{A}$ of Gaussian states, respectively).

The trace distance, arising (modulo a $\tfrac{1}{2}$ normalisation factor) from the 1-norm induced distance D₁ (see table 2), has also been adopted to define geometric measures of QCs as in equation (45) [78, 80, 122–124]. Because D_p is only contractive for p = 1, the trace distance based geometric measure, alias 'trace distance discord', is the only one of its class that directly obeys requirements (ii), (iv), and (v), hence emerging as the most suitable choice out of all the Schatten D_p norm induced geometric measures. For pure states ${\left|| ,\psi ,\rangle \right.\rangle }_{{AB}}$, only when subsystem A is a qubit it has been shown in [80] that this measure of QCs reduces to the entanglement negativity [115, 116], i.e. 

$$\begin{eqnarray}&&{Q}_{A}^{{G}_{1}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})={E}^{N}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}}),\end{eqnarray} \tag{ 71 }$$

while the behaviour of ${Q}_{A}^{{G}_{1}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})$ and ${Q}_{{AB}}^{{G}_{1}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})$ for arbitrary pure states is not known, leaving the fulfilment of requirement (iii) as an open question in general. A closed formula for ${Q}_{A}^{{G}_{1}}({\rho }_{{AB}})$ on two-qubit states with X-shaped density matrices (X states) and two-qubit quantum–classical states was provided in [59], where also the optimisation for general two-qubit states was greatly simplified. It was shown in [80] that, if subsystem A is a qubit, then

$$\begin{eqnarray}&&{Q}_{A}^{{G}_{1}}({\rho }_{{AB}})={Q}_{A}^{{M}_{1}}({\rho }_{{AB}})={Q}_{A}^{{E}^{N}}({\rho }_{{AB}})=\displaystyle \frac{1}{2}{Q}_{A}^{{U}_{1}}({\rho }_{{AB}}).\end{eqnarray} \tag{ 72 }$$

Operationally, the trace distance is related to the minimum probability of error in the discrimination between states [125]. Experimental investigations of ${Q}_{A}^{{G}_{1}}(\rho )$ have been carried out in a two-qubit nuclear magnetic resonance setup [104, 126].

The squared Bures distance ${D}_{\mathrm{Bu}}$, as well as the monotonically related infidelity D_F (see table 2), can induce another pair of geometric measures of QCs defined as in equations (45), that satisfy all of the five requirements of section 3.1 [27, 35, 127–129]. Indeed, ${D}_{\mathrm{Bu}}$ is contractive, and for pure states ${\left|| ,\psi ,\rangle \right.\rangle }_{{AB}}$ it holds that

$$\begin{eqnarray}&&{Q}_{A}^{{G}_{\mathrm{Bu}}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})={Q}_{{AB}}^{{G}_{\mathrm{Bu}}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})={E}^{{G}_{\mathrm{Bu}}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})=2(1-\sqrt{{s}_{\max }}),\end{eqnarray} \tag{ 73 }$$

where s_max is the maximum Schmidt coefficient of ${\left|| ,\psi ,\rangle \right.\rangle }_{{AB}}$ [127], which defines the geometric measure of entanglement [130]. An operational interpretation of ${Q}_{A}^{{G}_{\mathrm{Bu}}}({\rho }_{{AB}})$ can be provided in terms of the optimal success probability of an ambiguous quantum state discrimination task [127], as discussed later in section 4.2.2. Closed formulae for ${Q}_{A}^{{G}_{\mathrm{Bu}}}({\rho }_{{AB}})$ have been supplied for Bell diagonal states of two qubits [27, 128].

Similarly, using the squared Hellinger distance ${D}_{\mathrm{He}}$ (see table 2) one obtains as well valid measures of geometric QCs obeying all the requirements. An in-depth study of ${Q}_{A}^{{G}_{\mathrm{He}}}({\rho }_{{AB}})$ was carried out in [56], where they simplified the problem of evaluating this measure for arbitrary bipartite states, and showed in particular that for pure states ${\left|| ,\psi ,\rangle \right.\rangle }_{{AB}}$ it holds

$$\begin{eqnarray}&&{Q}_{A}^{{G}_{\mathrm{He}}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})=2\left((,1-\sqrt{{\rm{Tr}}({\rho }_{A}^{2})},)\right)=2\left((,1-\sqrt{{\rm{Tr}}({\rho }_{B}^{2})},)\right),\end{eqnarray} \tag{ 74 }$$

which is a measure of entanglement [62]. An interesting link to the Hilbert–Schmidt based geometric measure was also established [56],

$$\begin{eqnarray}&&{Q}_{A}^{{G}_{\mathrm{He}}}({\rho }_{{AB}})=2-2\sqrt{1-{Q}_{A}^{{G}_{2}}\left((,\sqrt{{\rho }_{{AB}}},)\right)},\end{eqnarray} \tag{ 75 }$$

which shows that, without compromising its reliability, the squared Hellinger distance based geometric measure maintains the same computational simplicity of the Hilbert–Schmidt distance based geometric measure corrected as in [113], and is thus amenable to a closed analytical formula for all bipartite states ${\rho }_{{AB}}$ when subsystem A is a qubit. Geometric measures of QCs based on the Hellinger distance have been discussed for Gaussian states in [131]. Operationally, the Hellinger distance admits an interpretation in terms of asymptotic state discrimination [56, 132], as detailed in section 4.2.2.

It is also interesting to compare the various geometric measures of QCs arising from different choices of distance D_δ. We can use general inequalities between the distances in table 2 to infer inequalities between the corresponding geometric measures, i.e. if for two choices of distance ${\delta }_{1}$ and ${\delta }_{2}$ it holds that $f({D}_{{\delta }_{1}}(\rho ,\sigma ))\leqslant g({D}_{{\delta }_{2}}(\rho ,\sigma ))$ for any ρ and σ and for two increasing functions f and g, then we know that

$$\begin{eqnarray}&&f\left((,{Q}_{A}^{{G}_{{\delta }_{1}}}({\rho }_{{AB}}),)\right)\leqslant g\left((,{Q}_{A}^{{G}_{{\delta }_{2}}}({\rho }_{{AB}}),)\right),\quad {\rm{and}}\quad f\left((,{Q}_{{AB}}^{{G}_{{\delta }_{1}}}({\rho }_{{AB}}),)\right)\leqslant g\left((,{Q}_{{AB}}^{{G}_{{\delta }_{2}}}({\rho }_{{AB}}),)\right).\end{eqnarray} \tag{ 76 }$$

The following inequalities are therefore useful [2, 56, 77, 79, 127, 133]

$$\begin{eqnarray} & & \displaystyle \frac{1}{r}{D}_{1}{\left((,\rho ,\sigma ,)\right)}^{2}\leqslant {D}_{2}(\rho ,\sigma )\leqslant {D}_{1}{\left((,\rho ,\sigma ,)\right)}^{2},\\ & & \displaystyle \frac{1}{2}{D}_{1}{\left((,\rho ,\sigma ,)\right)}^{2}\leqslant {D}_{\mathrm{RE}}(\rho ,\sigma ),\\ & & {D}_{\mathrm{Bu}}(\rho ,\sigma )\leqslant {D}_{\mathrm{He}}(\rho ,\sigma )\leqslant {D}_{1}(\rho ,\sigma )\leqslant \sqrt{1-{\left(1-\displaystyle \frac{1}{2}{D}_{\mathrm{Bu}}(\rho ,\sigma )\right)}^{2}},\end{eqnarray} \tag{ 77 }$$

where r is the rank of $\rho -\sigma$. These bounds are also applicable to the other types of measures of QCs involving a distance. One can further derive bounds between geometric measures of QCs that do not stem from fundamental inequalities between the corresponding distances. More generally, bounds amongst the different types of measures of QCs can be found, but these are sometimes tailored to the dimension of the relevant subsystems. A wealth of such inequalities are provided in [56].

There are other possible choices of ${D}_{\delta }$ in equation (45) that can be used to define geometric measures of QCs. First of all, note that both the Bures distance and Hellinger distance are closely related, in fact they become identical when considering the distance between pairs of commuting states. Indeed, both are examples of geodesic distances belonging to a family of monotone and Riemannian metrics defined on the set of quantum states, characterised by the Morozova–Čentsov–Petz theorem [77, 134, 135]. These infinitely many monotone and Riemannian metrics can each give a unique contractive geodesic distance that could in principle be used to define a geometric measure of QCs, but finding the geodesic distance in each case is a formidable problem and the information theoretic relevance of each distance would need to be considered.

Generalisations of the relative entropy could also be used to define geometric measures of QCs. Two recently suggested generalisations are the sandwiched relative Rényi entropies [136, 137] (see also [138])

$$\begin{eqnarray}&&{D}_{{{ \mathcal R }}_{\alpha }}(\rho ,\sigma )=\displaystyle \frac{\mathrm{log}{\rm{Tr}}\left[{({\sigma }^{\tfrac{1-\alpha }{2\alpha }}\rho {\sigma }^{\tfrac{1-\alpha }{2\alpha }})}^{\alpha }\right]}{\alpha -1}\end{eqnarray} \tag{ 78 }$$

and the sandwiched relative Tsallis entropies [139]

$$\begin{eqnarray}&&{D}_{{{ \mathcal T }}_{\alpha }}(\rho ,\sigma )=\displaystyle \frac{{\rm{Tr}}\left[{({\sigma }^{\tfrac{1-\alpha }{2\alpha }}\rho {\sigma }^{\tfrac{1-\alpha }{2\alpha }})}^{\alpha }\right]-1}{\alpha -1}.\end{eqnarray} \tag{ 79 }$$

These recent variations of the traditional relative Rényi and Tsallis entropies, designed to capture the noncommutative nature of the density matrix, are contractive quasi-distances for $\alpha \in (1/2,\infty ]$ and thus any geometric measure of QCs based upon either choice would naturally obey requirements (i), (ii), (iv) and (v), with requirement (iii) needing to be tested for each α. Certain values of α recover well known relative entropies, in particular when $\alpha \to 1$ we have

$$\begin{eqnarray}&&{D}_{{{ \mathcal R }}_{1}}(\rho ,\sigma )={D}_{{{ \mathcal T }}_{1}}(\rho ,\sigma )={D}_{\mathrm{RE}}(\rho ,\sigma ).\end{eqnarray} \tag{ 80 }$$

The measurement induced geometric type of measure was first introduced in [140] for the squared Hilbert–Schmidt distance and has subsequently been primarily studied in parallel with the geometric measures. In particular, due to the link in equation (67), ${Q}_{A}^{{M}_{2}}({\rho }_{{AB}})$ is not a valid measure of QCs as it fails requirement (iv), and similar inconsistencies can be expected for ${Q}_{{AB}}^{{M}_{2}}({\rho }_{{AB}})$ due to the lack of contractivity of the Hilbert–Schmidt distance D₂.

While we have coincidence between the one-sided geometric measures and the one-sided measurement induced geometric measures of QCs based on the trace distance for a qubit [80], as expressed in equation (72), it is still a relevant question to ask whether ${Q}_{A}^{{M}_{1}}({\rho }_{{AB}})$ and also ${Q}_{{AB}}^{{M}_{1}}({\rho }_{{AB}})$ are valid measures of QCs in general. As we have discussed in section 3.2.2, requirement (i) is naturally obeyed, and the contractivity of D₁ implies requirements (ii) and (iv), however it is not known whether requirement (v) holds. Regarding requirement (iii), it was proven in [141] by showing that ${Q}_{A}^{{M}_{1}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})={Q}_{{AB}}^{{M}_{1}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})$ and both amount to a measure of entanglement.

Two measurement induced geometric measures of QCs that do not have such a close link with the corresponding geometric measures are the ones induced by the squared Bures distance and the squared Hellinger distance (see table 2). The one-sided versions, considered in [56], are valid measures of QCs obeying all the requirements (i)–(iv), while requirement (v) has not been tested yet. In particular, for pure states ${\left|| ,\psi ,\rangle \right.\rangle }_{{AB}}$ it holds

$$\begin{eqnarray}&&{Q}_{A}^{{M}_{\mathrm{Bu}}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})={Q}_{A}^{{G}_{\mathrm{He}}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})=2\left((,1-\sqrt{{\rm{Tr}}({\rho }_{A}^{2})},)\right)=2\left((,1-\sqrt{{\rm{Tr}}({\rho }_{B}^{2})},)\right),\end{eqnarray} \tag{ 81 }$$

$$\begin{eqnarray}&&{Q}_{A}^{{M}_{\mathrm{He}}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})=2\left((,1-{\rm{Tr}}({\rho }_{A}^{3/2}),)\right)=2\left((,1-{\rm{Tr}}({\rho }_{B}^{3/2}),)\right),\end{eqnarray} \tag{ 82 }$$

which are measures of entanglement [62]. We point again to [56] for a compendium of informative bounds.

Finally, the extensions to other distances suggested in section 3.3.7 can likewise be considered for the measurement induced geometric measures.

A simple choice to gauge the change of information due to local measurements is the (negative) von Neumann entropy in table 3. By using ${I}_{{ \mathcal S }}=-{ \mathcal S }$ in equations (53) one can see that ${Q}_{A}^{{I}_{{ \mathcal S }}}({\rho }_{{AB}})$ and ${Q}_{{AB}}^{{I}_{{ \mathcal S }}}({\rho }_{{AB}})$ quantify the global information lost by (rank one) LPMs.

The measures ${Q}_{A}^{{I}_{{ \mathcal S }}}({\rho }_{{AB}})$ and ${Q}_{{AB}}^{{I}_{{ \mathcal S }}}({\rho }_{{AB}})$ were introduced in [87], being called the 'one-way quantum deficit' and 'zero-way quantum deficit', respectively. They have fundamental operational interpretations in thermodynamics, as we discuss in section 4.3.1. The coincidence between these measures and the relative entropy based geometric measures, equation (65), was highlighted in [87], and the coincidence with the distillable entanglement based entanglement activation measures, equation (66), was highlighted in [35, 36]. The one-sided measure ${Q}_{A}^{{I}_{{ \mathcal S }}}({\rho }_{{AB}})$ was also independently studied [94] and referred to as 'thermal discord' in [12]. All the requirements are satisfied for these measures as discussed in section 3.3.1.

The corresponding LGM based informational measures in terms of von Neumann entropy, ${Q}_{A}^{{\tilde{I}}_{{ \mathcal S }}}({\rho }_{{AB}})$ and ${Q}_{{AB}}^{{\tilde{I}}_{{ \mathcal S }}}({\rho }_{{AB}})$, were considered in [10, 142], with optimisation restricted to local rank one POVMs. These measures are less used in the literature and requirements (iv) and (v) were not tested, while requirement (iii) was proven in [142].

A final word of clarification is needed regarding the restriction to rank one measurements for von Neumann entropy based informational measures of QCs. Notice that, if one allowed instead the optimisation over all LGMs (or even all LPMs), then the optimal measurement could be the trivial 'measurement' of not doing anything (i.e., the identity), resulting in ill-defined quantifiers. The restriction to rank one measurements is therefore necessary to arrive at informative (and non-negative) QCs measures, and is anyway an intuitive restriction since rank one measurements represent the most fine grained type of measurements [143].

Since we are interested in QCs, it seems appropriate to consider as an informational gauge the mutual information ${I}_{{ \mathcal I }}={ \mathcal I }$, which is itself a measure of correlations. Specifically, resulting from a generalisation of the classical mutual information to the quantum setting, the mutual information in table 3 can be seen as the canonical quantifier of the total correlations between subsystem A and B in a state ${\rho }_{{AB}}$. We can think of the total correlations remaining in the states ${\tilde{{\rm{\Pi }}}}_{A}[{\rho }_{{AB}}]$ or ${\tilde{{\rm{\Pi }}}}_{{AB}}[{\rho }_{{AB}}]$ after a (general) LGM on subsystem A or on both subsystems A and B as effectively classical, because these are the correlations that are extracted from ${\rho }_{{AB}}$ by a local measurement and can be stored in a classical register. Hence, to find the corresponding QCs one should subtract these classical correlations from the original total correlations of ${\rho }_{{AB}}$. Since different local measurements may lead to the extraction of different amounts of classical correlations, and given that we aim to quantify only the genuinely quantum share of the total correlations, we need to minimise this difference (or equivalently maximise the extractable classical correlations) over all LGMs, hence justifying the form of the mutual information based measures of QCs in equations (52) and (53). These measures will therefore be understood as quantifying, in general, the minimum amount of correlations lost in the state ${\rho }_{{AB}}$ by the act of a maximally informative local measurement on one or both subsystems.

It turns out that the one-sided mutual information based measure of QCs coincides with the original definition of 'quantum discord', which was first conceived in [6, 7] by interpreting the disagreement (or discord, indeed) between two possible extensions of classically identical definitions of the mutual information to the quantum setting. In [6] Ollivier and Zurek considered ${Q}_{A}^{{I}_{{ \mathcal I }}}({\rho }_{{AB}})$, i.e., minimisation over LPMs, while in [7] Henderson and Vedral considered ${Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}})$, i.e., minimisation over LGMs. Having clarified the fundamental meaning of quantum discord within the broader scope of the informational approach to QCs, for completeness we will now provide a summary of the original analysis behind it by focusing on the more general case of LGMs, although the same holds analogously for LPMs.

Consider two classical random variables A and B with a joint probability distribution p_ab and marginal probability distributions ${p}_{a}={\sum }_{b\in B}{p}_{{ab}}$ and ${p}_{b}={\sum }_{a\in A}{p}_{{ab}}$. The correlations between A and B are given by the mutual information

$$\begin{eqnarray}&&{ \mathcal I }(A\,:B):= { \mathcal H }(A)+{ \mathcal H }(B)-{ \mathcal H }({AB}),\end{eqnarray} \tag{ 83 }$$

with ${ \mathcal H }(A)=-{\sum }_{a\in A}{p}_{a}\mathrm{log}{p}_{a}$ and ${ \mathcal H }(B)=-{\sum }_{b\in B}{p}_{b}\mathrm{log}{p}_{b}$ the Shannon entropies of the individual random variables A and B, and ${ \mathcal H }({AB})=-{\sum }_{a\in A}{\sum }_{b\in B}{p}_{{ab}}\mathrm{log}{p}_{{ab}}$ the joint Shannon entropy of A and B [144, 145]. An equivalent expression for the mutual information is given by

$$\begin{eqnarray}&&{ \mathcal J }(A\,:B):= { \mathcal H }(B)-{ \mathcal H }(B| A),\end{eqnarray} \tag{ 84 }$$

where

$$\begin{eqnarray}&&{ \mathcal H }(B| A):= \displaystyle \sum _{a\in A}\,{p}_{a}{ \mathcal H }(B| a)\end{eqnarray} \tag{ 85 }$$

is the conditional Shannon entropy, i.e. the average of ${ \mathcal H }(B| a)=-{\sum }_{b\in B}{p}_{b| a}\mathrm{log}{p}_{b| a}$ over $a\in A$, with ${p}_{b| a}:= {p}_{{ab}}/{p}_{a}$ the conditional probability distribution of B given result a. By substituting the definition of ${p}_{b| a}$ into the above expression of ${ \mathcal H }(B| A)$, we can easily see that

$$\begin{eqnarray}&&{ \mathcal H }(B| A)={ \mathcal H }({AB})-{ \mathcal H }(A),\end{eqnarray} \tag{ 86 }$$

establishing the equivalence of ${ \mathcal I }(A\,:B)$ and ${ \mathcal J }(A\,:B)$.

By extending the above to the quantum setting, the joint probability distribution p_ab generalises to a state ${\rho }_{{AB}}$ of the composite system and the marginal probability distributions p_a and p_b become states of the respective subsystems, i.e. ${\rho }_{A}={{\rm{Tr}}}_{B}(\rho )$ and ${\rho }_{B}={{\rm{Tr}}}_{A}(\rho )$. The first definition ${ \mathcal I }(A\,:B)$ of the mutual information then naturally extends to

$$\begin{eqnarray}&&{ \mathcal I }({\rho }_{{AB}}):= { \mathcal S }({\rho }_{A})+{ \mathcal S }({\rho }_{B})-{ \mathcal S }({\rho }_{{AB}}),\end{eqnarray} \tag{ 87 }$$

with the von Neumann entropy ${ \mathcal S }$ replacing the Shannon entropy. Extending the second definition ${ \mathcal J }(A\,:B)$ of the mutual information is less simple because it is not unambiguously clear how to deal with the conditional entropy ${ \mathcal H }(B| A)$. The most direct way would be to emulate equation (86) with the conditional entropy

$$\begin{eqnarray}&&{{ \mathcal S }}_{B| A}({\rho }_{{AB}}):= { \mathcal S }({\rho }_{{AB}})-{ \mathcal S }({\rho }_{A}).\end{eqnarray} \tag{ 88 }$$

However, to truly capture the conditioning of 'B given A' in the quantum setting, one must first extract information on A by performing local measurements. Hence, following equation (85), the conditional entropy can be interpreted as the average entropy of the post-measurement ensemble of states of subsystem B following an LGM on A with subnormalised maps $\{{({\tilde{\pi }}_{A})}_{a}\}$. Invoking the definitions in section 2.3.1, one can then define the LGM based conditional von Neumann entropy as

$$\begin{eqnarray}&&{{ \mathcal S }}_{B| A}^{{\tilde{\pi }}_{A}}({\rho }_{{AB}}):= \displaystyle \sum _{a}\,{p}_{a}{ \mathcal S }({\rho }_{B| a}),\end{eqnarray} \tag{ 89 }$$

which gives an LGM based quantum version of the classical mutual information as

$$\begin{eqnarray}&&{\tilde{{ \mathcal J }}}_{A}^{{\tilde{\pi }}_{A}}({\rho }_{{AB}}):= { \mathcal S }({\rho }_{B})-{{ \mathcal S }}_{B| A}^{{\tilde{\pi }}_{A}}({\rho }_{{AB}}).\end{eqnarray} \tag{ 90 }$$

As argued before, ${ \mathcal I }({\rho }_{{AB}})$ in equation (87) represents the total correlations, while the maximisation of the quantity in equation (90) over all LGMs,

$$\begin{eqnarray}&&{\tilde{{ \mathcal J }}}_{A}({\rho }_{{AB}}):= \mathop{\sup }\limits_{{\tilde{\pi }}_{A}}\left[[,{\tilde{{ \mathcal J }}}_{A}^{{\tilde{\pi }}_{A}}({\rho }_{{AB}}),]\right],\end{eqnarray} \tag{ 91 }$$

can be regarded as a measure of one-sided classical correlations (with respect to subsystem A). Their difference yields the (LGM based) quantum discord with respect to subsystem A,

$$\begin{eqnarray*}&&{\tilde{{ \mathcal D }}}_{A}({\rho }_{{AB}}):= { \mathcal I }({\rho }_{{AB}})-{\tilde{{ \mathcal J }}}_{A}({\rho }_{{AB}})=\mathop{\inf }\limits_{{\tilde{\pi }}_{A}}\left[[,{ \mathcal I }({\rho }_{{AB}})-{\tilde{{ \mathcal J }}}_{A}^{{\tilde{\pi }}_{A}}({\rho }_{{AB}}),]\right],\end{eqnarray*}$$

which may be equivalently seen as the difference between two quantum versions of the classical conditional entropy

$$\begin{eqnarray}&&{\tilde{{ \mathcal D }}}_{A}({\rho }_{{AB}})=\mathop{\inf }\limits_{{\tilde{\pi }}_{A}}\left[[,{{ \mathcal S }}_{B| A}^{{\tilde{\pi }}_{A}}({\rho }_{{AB}})-{{ \mathcal S }}_{B| A}({\rho }_{{AB}}),]\right].\end{eqnarray} \tag{ 92 }$$

To see that this quantity coincides with the informational QCs measure that we label ${Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}})$, as defined in equation (52), we can write explicitly (see e.g. [33])

$$\begin{eqnarray}\begin{array}{rcl}{\tilde{{ \mathcal J }}}_{A}^{{\tilde{\pi }}_{A}}({\rho }_{{AB}}) & = & { \mathcal S }({\rho }_{B})-{{ \mathcal S }}_{B| A}^{{\tilde{\pi }}_{A}}({\rho }_{{AB}})={ \mathcal S }({\rho }_{B})-\displaystyle \sum _{a}\,{p}_{a}{ \mathcal S }({\rho }_{B| a})\\ & = & { \mathcal S }({\rho }_{B})-\displaystyle \sum _{a}\,{p}_{a}\mathrm{log}{p}_{a}-{ \mathcal S }\left(\displaystyle \sum _{a}\,{p}_{a}\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{{A}^{\prime }}\otimes {\rho }_{B| a}\right)\\ & = & { \mathcal I }\left(\displaystyle \sum _{a}\,{p}_{a}\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{{A}^{\prime }}\otimes {\rho }_{B| a}\right)={ \mathcal I }\left((,{\tilde{{\rm{\Pi }}}}_{A}\left[[,{\rho }_{{AB}},]\right],)\right),\end{array}\end{eqnarray} \tag{ 93 }$$

where in the second line we use the joint entropy theorem [2], and in the last we use equation (16).

The two-sided informational measure based upon mutual information has also been extensively studied in the literature. In [97], a modification of ${Q}_{{AB}}^{{I}_{{ \mathcal I }}}({\rho }_{{AB}})$ was considered by omitting the optimisation over local measurements and called the 'measurement induced disturbance'. Local projections fixed in the respective eigenbases of the reduced states were used instead in its definition, i.e. if ${\rho }_{A}={\sum }_{a}{p}_{a}\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{A}$ and ${\rho }_{B}={\sum }_{b}{p}_{b}\left|| ,b,\rangle \right.\rangle {\left.\langle \langle ,b,| \right|}_{B}$ then the chosen LPM would consist of projectors $\{{({\pi }_{A})}_{a}\otimes {({\pi }_{A})}_{b}\}$ with ${({\pi }_{A})}_{a}[\rho ]=\left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{A}\rho \left|| ,a,\rangle \right.\rangle {\left.\langle \langle ,a,| \right|}_{A}$ and ${({\pi }_{B})}_{b}[\rho ]=\left|| ,b,\rangle \right.\rangle {\left.\langle \langle ,b,| \right|}_{B}\rho \left|| ,b,\rangle \right.\rangle {\left.\langle \langle ,b,| \right|}_{B}$. The motivation behind this choice is so that the local projection leaves the reduced states of ${\rho }_{{AB}}$ invariant, simplifying the measurement induced disturbance to the difference in global von Neumann entropies and thus leading to an upper bound to ${Q}_{{AB}}^{{I}_{{ \mathcal S }}}({\rho }_{{AB}})$. However, ignoring the optimisation over local measurements generally causes an overestimation of the QCs and also gives rise to pathologies such as the measurement induced disturbance being undefined on states whose marginals have a degenerate spectrum [143], or even approaching a maximum on some classical–classical states where QCs vanish instead [146]. To overcome this problem, the measure ${Q}_{{AB}}^{{I}_{{ \mathcal I }}}({\rho }_{{AB}})$ including an optimisation over LPMs on A and B was studied in [146] as the 'ameliorated measurement induced disturbance', while the more general LGM based measure ${Q}_{{AB}}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}})$, sometimes referred to as 'symmetric quantum discord' had been considered already in [24, 143]. In particular, the authors of [147–149] defined the (two-sided) 'classical mutual information'

$$\begin{eqnarray}&&{\tilde{{ \mathcal J }}}_{{AB}}({\rho }_{{AB}}):= \mathop{\sup }\limits_{{\tilde{{\rm{\Pi }}}}_{{AB}}}{ \mathcal I }\left((,{\tilde{{\rm{\Pi }}}}_{{AB}}[{\rho }_{{AB}}],)\right),\end{eqnarray} \tag{ 94 }$$

as the maximum classical correlations obtainable from a quantum state ${\rho }_{{AB}}$ by purely local processing. The quantity in equation (94) amounts to the generalisation of equation (91) when LGMs ${\tilde{{\rm{\Pi }}}}_{{AB}}={\tilde{{\rm{\Pi }}}}_{A}\otimes {\tilde{{\rm{\Pi }}}}_{B}$ on both A and B are considered as in equation (17). In this way, the two-sided mutual information based measure of QCs can then be written simply as

$$\begin{eqnarray}&&{Q}_{{AB}}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}})={ \mathcal I }({\rho }_{{AB}})-{\tilde{{ \mathcal J }}}_{{AB}}({\rho }_{{AB}}),\end{eqnarray} \tag{ 95 }$$

that is, as the difference between total and classical mutual information, in analogy to equation (92).

Non-negativity of ${Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}})$ follows directly from the strong subadditivity of the von Neumann entropy [150] (see also [6, 29, 151]). This implies non-negativity of all the other mutual information based informational measures of QCs, see equation (98) in the following for clarification.

As all informational measures, requirements (i) and (ii) are satisfied. Furthermore, requirement (iv) was proven to hold in [35, 111], and requirement (iii) holds as well, since for pure states ${\left|| ,\psi ,\rangle \right.\rangle }_{{AB}}$ we have  [33, 77, 142]

$$\begin{eqnarray}&&{Q}_{A}^{{I}_{{ \mathcal I }}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})={Q}_{{AB}}^{{I}_{{ \mathcal I }}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})={Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})={Q}_{{AB}}^{{\tilde{I}}_{{ \mathcal I }}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})={ \mathcal S }({\rho }_{A})={ \mathcal S }({\rho }_{B}),\end{eqnarray} \tag{ 96 }$$

which is the entanglement entropy. However, it is still an open question whether requirement (v) is obeyed for this class of measures.

Regarding computability, the main obstacle in calculation of the mutual information based informational measures of QCs is the non-trivial optimisation over LGMs or LPMs. Indeed, the computational complexity of this optimisation is NP-complete [152], falling into the hardest class of problems within NP and with no efficient way currently known to provide a solution, so that the running time of any present algorithm grows exponentially with the dimension of the Hilbert space. Nevertheless, in the LGM case it is sufficient to optimise only over local POVMs that are extremal and of rank one (see section 2.3.1), as proven in [151, 153, 154]. Notice that, unlike the case of the von Neumann ${I}_{{ \mathcal S }}$ based informational measures in section 3.5.1, the mutual information ${I}_{{ \mathcal I }}$ based measures defined in this section can be optimised in principle over all LGMs or LPMs, without the need to restrict their definitions a priori, yet the optimal measurements always happen to consist of rank one operators [142, 143, 151].

Comparisons between minimisation over LGMs and LPMs have been carried out, in particular for the quantum discord, and even for two-qubit states there are many cases where ${Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}})\lt {Q}_{A}^{{I}_{{ \mathcal I }}}({\rho }_{{AB}})$, although the two quantities are typically quite close [142, 155–157]. Nevertheless, there has been some analytical progress in the calculation of these quantities, including: the evaluation of ${Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}})$ in [158] for all two-qubit states of rank 2, and in [98, 156, 157, 159–162] for certain two-qubit X states (note that the original result of [159] was claimed to apply to all two-qubit X states, but this claim was later shown to be incorrect in [156, 160, 163]), and in [98] for highly symmetric two-qudit states; and the evaluation of ${Q}_{A}^{{I}_{{ \mathcal I }}}({\rho }_{{AB}})$ in [97] for two-qubit Bell diagonal states (in which case ${Q}_{A}^{{I}_{{ \mathcal I }}}({\rho }_{{AB}})={Q}_{{AB}}^{{I}_{{ \mathcal I }}}({\rho }_{{AB}})$), in [164–167] for two-qubit X states, in [101] for qubit-qudit extended X states, in [168] for certain two-qutrit states, in [169, 170] for two-qubit reduced states of ground states of spin chains, and in [171, 172] by providing an efficient numerical algorithm for general two-qubit states. The quantum discord ${Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}})$ has been generalised to continuous variable Gaussian states in [120, 173], and a closed computable formula obtained for all two-mode Gaussian states with optimisation restricted to Gaussian LGMs [120]; the latter measurements turn out to be in fact optimal (even among non-Gaussian measurements) for a large class of two-mode Gaussian states [174]. Two-sided measures of QCs based on the mutual information ${Q}_{{AB}}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}})$ have also been studied for Gaussian states [175], showing that in this case Gaussian LGMs are generally not optimal.

We will detail some of the most concrete and direct operational interpretations of the quantum discord and other mutual information based informational measures of QCs in section 4. We will also discuss the curious connection between ${Q}_{A}^{{I}_{{ \mathcal I }}}({\rho }_{{AB}})$ and the entanglement activation type measures in section 3.6. Finally, let us mention a useful inequality involving the quantum discord ${Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}$ and the entanglement of formation E^f (see table 4), in arbitrary tripartite states ${\rho }_{{ABC}}$ (with equality on pure states), given by [176, 177]

$$\begin{eqnarray}&&{Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}})\geqslant { \mathcal S }({\rho }_{A})-{ \mathcal S }({\rho }_{{AB}})+{E}_{B:C}^{f}({\rho }_{{BC}}).\end{eqnarray} \tag{ 97 }$$

The following wheel of inequalities summarises the hierarchical structure of the informational measures of QCs introduced so far, when all evaluated on the same arbitrary bipartite state ${\rho }_{{AB}}$:

The vertical inequalities arise because an optimisation over LGMs necessarily includes an optimisation over LPMs. The horizontal inequalities are shown in [87, 143]. Finally, the radial inequalities are explored in [142].

From an information theoretic perspective, it is often relevant to consider also an asymptotic scenario in which many independent and identically distributed (i.i.d.) copies of a quantum state are available. Given a real, non-negative function $f(\rho )$ on the set of quantum states, the corresponding regularisation (if the limit exists) is defined as

$$\begin{eqnarray}&&f{\left((,\rho ,)\right)}_{\infty }:= \mathop{\mathrm{lim}}\limits_{N\to \infty }\displaystyle \frac{1}{N}\,f({\rho }^{\otimes N}).\end{eqnarray} \tag{ 99 }$$

Regularisation has been used extensively in entanglement theory [3], in particular to define additive entanglement measures (like the entanglement cost) from non-additive ones (like the entanglement of formation), since for any $f(\rho )$ the regularised $f{(\rho )}_{\infty }$ is by construction additive on tensor product states.

One can then define accordingly regularised versions of all of the measures of QCs mentioned in this review. In particular, for the informational type of measures, a remarkable convergence occurs upon regularisation, collapsing the whole left half of the hierarchy in equation (98). Specifically, it was shown in [178] that the regularisations of LGM and LPM versions of both the mutual information and the von Neuman entropy based informational QCs measures all reduce to the same quantity, i.e. 

$$\begin{eqnarray}&&{Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}{\left((,{\rho }_{{AB}},)\right)}_{\infty }={Q}_{A}^{{I}_{{ \mathcal I }}}{\left((,{\rho }_{{AB}},)\right)}_{\infty }={Q}_{A}^{{\tilde{I}}_{{ \mathcal S }}}{\left((,{\rho }_{{AB}},)\right)}_{\infty }={Q}_{A}^{{I}_{{ \mathcal S }}}{\left((,{\rho }_{{AB}},)\right)}_{\infty }.\end{eqnarray} \tag{ 100 }$$

This means in particular that the regularised one-way information deficit and the regularised one-sided quantum discord coincide in general. The quantity in equation (100) can equivalently be written as

$$\begin{eqnarray}&&{Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}{\left((,{\rho }_{{AB}},)\right)}_{\infty }={ \mathcal I }({\rho }_{{AB}})-{\tilde{{ \mathcal J }}}_{A}{\left((,{\rho }_{{AB}},)\right)}_{\infty },\end{eqnarray} \tag{ 101 }$$

which is obtained from equation (92) upon regularisation, taking into account that the mutual information is already additive, ${ \mathcal I }{({\rho }_{{AB}})}_{\infty }={ \mathcal I }({\rho }_{{AB}})$. The complementary quantity ${\tilde{{ \mathcal J }}}_{A}{({\rho }_{{AB}})}_{\infty }$ in equation (100), which is the regularised version of the one-sided measure of classical correlations [7] defined in equation (91), is known as 'distillable common randomness' [10, 179].

There are natural extensions of the two informational measures suggested in table 3. Generalisations of the von Neumann entropy are considered in [180], in particular the Rényi entropies

$$\begin{eqnarray}&&{{ \mathcal R }}_{\alpha }(\rho ):= -\displaystyle \frac{\mathrm{log}{\rm{Tr}}\left((,{\rho }^{\alpha },)\right)}{\alpha -1}\end{eqnarray} \tag{ 102 }$$

and the Tsallis entropies

$$\begin{eqnarray}&&{{ \mathcal T }}_{\alpha }(\rho ):= \displaystyle \frac{1-{\rm{Tr}}\left((,{\rho }^{\alpha },)\right)}{\alpha -1},\end{eqnarray} \tag{ 103 }$$

with, as $\alpha \to 1$,

$$\begin{eqnarray}&&{{ \mathcal R }}_{1}(\rho )={{ \mathcal T }}_{1}(\rho )={ \mathcal S }(\rho ).\end{eqnarray} \tag{ 104 }$$

These generalised entropies can give extensions of the von Neumann entropy based informational measures of QCs by choosing in equation (53) the informational quantities ${I}_{{{ \mathcal R }}_{\alpha }}=-{{ \mathcal R }}_{\alpha }$ and ${I}_{{{ \mathcal T }}_{\alpha }}=-{{ \mathcal T }}_{\alpha }$. Indeed, in [180, 181] the authors consider ${Q}_{A}^{{I}_{{{ \mathcal R }}_{\alpha }}}({\rho }_{{AB}})$, ${Q}_{{AB}}^{{I}_{{{ \mathcal R }}_{\alpha }}}({\rho }_{{AB}})$, ${Q}_{A}^{{I}_{{{ \mathcal T }}_{\alpha }}}({\rho }_{{AB}})$ and ${Q}_{{AB}}^{{I}_{{{ \mathcal T }}_{\alpha }}}({\rho }_{{AB}})$, and even more general forms of entropy. These quantities are all non-negative because a complete rank one LPM always increases the Rényi and Tsallis entropies when the result is not known. Furthermore, ${I}_{{{ \mathcal R }}_{\alpha }}({\rho }_{{AB}})$ and ${I}_{{{ \mathcal T }}_{\alpha }}({\rho }_{{AB}})$ are invariant under local unitaries and hence requirements (i) and (ii) hold. For pure states ${\left|| ,\psi ,\rangle \right.\rangle }_{{AB}}$, one has

$$\begin{eqnarray}&&{Q}_{A}^{{I}_{{{ \mathcal R }}_{\alpha }}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})={Q}_{{AB}}^{{I}_{{{ \mathcal R }}_{\alpha }}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})={{ \mathcal R }}_{\alpha }({\rho }_{A})={{ \mathcal R }}_{\alpha }({\rho }_{B}),\end{eqnarray} \tag{ 105 }$$

$$\begin{eqnarray}&&{Q}_{A}^{{I}_{{{ \mathcal T }}_{\alpha }}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})={Q}_{{AB}}^{{I}_{{{ \mathcal T }}_{\alpha }}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})={{ \mathcal T }}_{\alpha }({\rho }_{A})={{ \mathcal T }}_{\alpha }({\rho }_{B}),\end{eqnarray} \tag{ 106 }$$

which are both valid generalisations of the entanglement entropy [77], thus validating requirement (iii). However, requirement (iv) does not hold for the family of informational measures corresponding to the Tsallis entropies for any choice of $\alpha \ne 1$ [182], because these measures can arbitrarily change with the addition and removal of an impure ancilla, as in equation (69). Indeed, when $\alpha =2$ the Tsallis entropies reduces to the linear entropy ${{ \mathcal T }}_{2}(\rho )=1-{\rm{Tr}}({\rho }^{2})$ and it holds that [183]

$$\begin{eqnarray}&&{Q}_{A}^{{I}_{{{ \mathcal T }}_{2}}}({\rho }_{{AB}})={Q}_{A}^{{M}_{2}}({\rho }_{{AB}})={Q}_{A}^{{G}_{2}}({\rho }_{{AB}}),\end{eqnarray} \tag{ 107 }$$

which is the Hilbert–Schmidt based geometric measure of QCs. It was then suggested in [182] to rescale these measures by a type of generalised purity, analogous to the suggestion of [96], but it is still not known whether requirement (iv) may be recovered in this way. The family of informational measures corresponding to the Rényi entropies are left unchanged by the addition and removal of an impure ancilla [184], but it is still unknown whether the more general requirement (iv) holds. Finally, it is not known whether requirement (v) is obeyed either.

One can also look at generalisations of the mutual information adopted as informational quantifier ${I}_{{ \mathcal I }}$. Recall that the mutual information ${ \mathcal I }({\rho }_{{AB}})$, defined in equation (87), can be equivalently rewritten as the distance from ${\rho }_{{AB}}$ to the set of product states according to the relative entropy [89, 93], i.e. 

$$\begin{eqnarray}&&{ \mathcal I }({\rho }_{{AB}})=\mathop{\inf }\limits_{{\sigma }_{{AB}}\in {P}_{{AB}}}{D}_{\mathrm{RE}}({\rho }_{{AB}},{\sigma }_{{AB}}),\end{eqnarray} \tag{ 108 }$$

where ${P}_{{AB}}$ is the set of product states, and the minimisation is achieved in general by ${\sigma }_{{AB}}={\rho }_{A}\otimes {\rho }_{B}$, the product of the marginals of ${\rho }_{{AB}}$. One could instead consider in equation (53) the informational quantities

$$\begin{eqnarray}{I}_{{{ \mathcal I }}_{\alpha }^{{ \mathcal R }}}({\rho }_{{AB}}) & = & \mathop{\inf }\limits_{{\chi }_{{AB}}\in { \mathcal P }}{D}_{{{ \mathcal R }}_{\alpha }}({\rho }_{{AB}},{\chi }_{{AB}}),\\ {I}_{{{ \mathcal I }}_{\alpha }^{{ \mathcal T }}}({\rho }_{{AB}}) & = & \mathop{\inf }\limits_{{\chi }_{{AB}}\in { \mathcal P }}{D}_{{{ \mathcal T }}_{\alpha }}({\rho }_{{AB}},{\chi }_{{AB}}),\end{eqnarray} \tag{ 109 }$$

arising from the sandwiched relative Rényi and Tsallis entropies defined in equations (78) and (79), to define generalised mutual information based measures of QCs ${Q}_{A}^{{I}_{{{ \mathcal I }}_{\alpha }^{{ \mathcal R }}}}({\rho }_{{AB}})$ and ${Q}_{A}^{{I}_{{{ \mathcal I }}_{\alpha }^{{ \mathcal T }}}}({\rho }_{{AB}})$ with optimisation over LPMs, as was done in [185]. The corresponding two-sided LPM and one-sided and two-sided LGM measures may be defined as well. Being non-negative, zero on classical states and local unitarily invariant, all these quantities obey requirements (i) and (ii). However, it is unknown whether requirements (iii), (iv), and (v) hold, and one may need to check in each case for particular values of α (beyond the standard $\alpha \to 1$).

We finally highlight that similar approaches to defining measures of QCs by generalising entropies have been reported [186–191].

Both [36] and [35] reached the conclusion that the entanglement activation measures of QCs based upon relative entropy ${E}^{\mathrm{RE}}$ and distillable entanglement E^d (see tables 2 and 4) coincide with each other and reduce to their geometric counterparts, see equation (66), and to the informational quantum deficits, as amply discussed in section 3.3.1. This is a consequence of the fact that the pre-measurement states as defined in the entanglement activation approach belong to the special class of so-called maximally correlated states, for which ${E}^{d}={E}^{\mathrm{RE}}$ [192]; more generally, this observation is crucial to establish requirement (i) for all the QCs measures ${Q}_{A}^{{E}^{\zeta }}$ and ${Q}_{{AB}}^{{E}_{\zeta }}$ even when using entanglement monotones ${E}^{\zeta }$ that may vanish on some non-separable states, such as the distillable entanglement E^d.

It is interesting to note that the entanglement activation approach can also be adapted to recover the original measure of quantum discord ${Q}_{A}^{{I}_{{ \mathcal I }}}({\rho }_{{AB}})$ optimised over LPMs, as anticipated in section 3.5.2 [35]. This requires a modification of the one-sided definition in equations (56), to consider the minimisation of the partial entanglement in the pre-measurement state,

$$\begin{eqnarray}&&{Q}_{A}^{{\rm{\Delta }}{E}^{\zeta }}({\rho }_{{AB}}):= \mathop{\inf }\limits_{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}\left[[,{E}_{{AB}:A^{\prime} }^{\zeta }\left((,\rho {^{\prime} }_{{ABA}^{\prime} }^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}},)\right)-{E}_{A:A^{\prime} }^{\zeta }\left((,{{\rm{Tr}}}_{B}\left((,\rho {^{\prime} }_{{ABA}^{\prime} }^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}},)\right),)\right),]\right],\end{eqnarray} \tag{ 110 }$$

that is, the difference given by the entanglement created between the whole system AB and the ancilla $A^{\prime}$, minus the entanglement created between the probed subsystem A and the ancilla $A^{\prime}$ in their reduced state, discarding the other subsystem B. It then turns out that [35]

$$\begin{eqnarray}&&{Q}_{A}^{{\rm{\Delta }}{E}^{\zeta }}({\rho }_{{AB}})={Q}_{A}^{{I}_{{ \mathcal I }}}({\rho }_{{AB}}),\end{eqnarray} \tag{ 111 }$$

with ${E}^{\zeta }$ being either ${E}^{\mathrm{RE}}$ or E^d, thus providing an alternative interpretation to the quantum discord [6].

Note that one could define more general measures of QCs ${Q}_{{AB}}^{{\rm{\Delta }}{E}^{\zeta }}$ based on the partial entanglement activation approach of equation (110) by considering other choices for ${E}^{\zeta }$, however in this case one needs ${E}^{\zeta }$ to be convex and non-increasing on average under stochastic LOCC (which are stronger requirements than standard LOCC monotonicity) [16], in order to generate valid measures of QCs, in particular to fulfil requirement (iv) [35].

The 'negativity of quantumness' was defined in [36] and studied in detail in [80]. One-sided and two-sided versions of this measure, ${Q}_{A}^{{E}^{N}}({\rho }_{{AB}})$ and ${Q}_{{AB}}^{{E}^{N}}({\rho }_{{AB}})$, are obtained by choosing the entanglement negativity E^N (see table 4) [115, 116] in equation (56). As already noticed in [36, 80], the negativity of quantumness amounts to the minimum coherence in all local bases quantified by the ℓ₁ norm (see table 5). We can formalise this equivalence using the notation of section 3.2.6. Namely, for a state ${\rho }_{{AB}}$ and fixing local orthonormal reference bases $\{{\left|| ,a,\rangle \right.\rangle }_{A}\}$ and $\{{\left|| ,b,\rangle \right.\rangle }_{B}\}$, let us define

$$\begin{eqnarray}{C}_{A}^{{{\ell }}_{1}\ \{{\left|| ,a,\rangle \right.\rangle }_{A}\}}({\rho }_{{AB}}) & := & \displaystyle \sum _{{ij}}\,{\parallel }_{A}\left.\langle \langle ,{a}_{i},| \right|{\rho }_{{AB}}\left|| ,{a}_{j},\rangle \right.\rangle {}_{A}\,{\parallel }_{1}-1,\\ {C}_{{AB}}^{{{\ell }}_{1}\ \{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}}({\rho }_{{AB}}) & := & \parallel {\rho }_{{AB}}{\parallel }_{{{\ell }}_{1}}^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\otimes {\left|| ,b,\rangle \right.\rangle }_{B}\}}-1,\end{eqnarray} \tag{ 112 }$$

where we have explicitly indexed the local basis $\{{\left|| ,a,\rangle \right.\rangle }_{A}\}\equiv {\{{\left|| ,{a}_{i},\rangle \right.\rangle }_{A}\}}_{i=1}^{{d}_{A}}$ in the one-sided case, while the notation in the two-sided case indicates that the ℓ₁ norm is taken for the matrix representation of ${\rho }_{{AB}}$ with respect to the product basis $\{\left|| ,a,\rangle \right.\rangle \otimes \left|| ,b,\}\right\}$ (recall that the ℓ₁ norm, or taxicab norm, is basis dependent). The minimisation of equation (112) over all local reference bases as in equation (61) defines then the one-sided and two-sided ℓ₁ coherence based measures of QCs, which coincide with the corresponding versions of the negativity of quantumness [80],

$$\begin{eqnarray}{Q}_{A}^{{C}^{{{\ell }}_{1}}}({\rho }_{{AB}}) & := & \mathop{\inf }\limits_{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}{C}_{A}^{{{\ell }}_{1}\ \{{\left|| ,a,\rangle \right.\rangle }_{A}\}}({\rho }_{{AB}})={Q}_{A}^{{E}^{N}}({\rho }_{{AB}}),\\ {Q}_{{AB}}^{{C}^{{{\ell }}_{1}}}({\rho }_{{AB}}) & := & \mathop{\inf }\limits_{\{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}}{C}_{{AB}}^{{{\ell }}_{1}\ \{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}}({\rho }_{{AB}})={Q}_{{AB}}^{{E}^{N}}({\rho }_{{AB}}).\end{eqnarray} \tag{ 113 }$$

The two-sided negativity of quantumness ${Q}_{{AB}}^{{E}^{N}}({\rho }_{{AB}})$ can also be interpreted as a geometric measure as in equation (45) and as a measurement induced geometric measure as in equation (50),

$$\begin{eqnarray}{Q}_{{AB}}^{{E}^{N}}({\rho }_{{AB}}) & = & {Q}_{{AB}}^{{G}_{{{\ell }}_{1}}}({\rho }_{{AB}})=\mathop{\inf }\limits_{{\chi }_{{AB}}\in {C}_{{AB}}}{D}_{{{\ell }}_{1}}({\rho }_{{AB}},{\chi }_{{AB}}),\\ {Q}_{{AB}}^{{E}^{N}}({\rho }_{{AB}}) & = & {Q}_{{AB}}^{{M}_{{{\ell }}_{1}}}({\rho }_{{AB}})=\mathop{\inf }\limits_{{\pi }_{{AB}}}{D}_{{{\ell }}_{1}}\left((,{\rho }_{{AB}},{\pi }_{{AB}}[{\rho }_{{AB}}],)\right),\end{eqnarray} \tag{ 114 }$$

where we adopt the ℓ₁ distance

$$\begin{eqnarray}&&{D}_{{{\ell }}_{1}}(\rho ,\sigma ):= \parallel \rho -\sigma {\parallel }_{{{\ell }}_{1}}^{\{\left|| ,i,\rangle \right.\rangle \}},\end{eqnarray} \tag{ 115 }$$

evaluated with respect to the basis $\{{\left|| ,a,\rangle \right.\rangle }_{A}\otimes {\left|| ,b,\rangle \right.\rangle }_{B}\}$, which corresponds in the first relation to the eigenbasis of the classical–classical state ${\chi }_{{AB}}$, and in the second relation to the product of local bases associated with the complete rank one LPM as defined in equation (19).

In the special case of subsystem A being a qubit, the one-sided negativity of quantumness ${Q}_{{AB}}^{{E}^{N}}({\rho }_{{AB}})$ relates instead to the geometric, measurement induced, and unitary response measures based on the trace distance [80], as summarised in equation (72).

Closed formulae for the negativity of quantumness of two-qubit Bell diagonal states and two-qudit highly symmetric states were obtained in [80]. Two optical experiments recently investigated the entanglement activation framework [38, 193]. In particular, in [38] a procedure was devised and tested to verify the 'if and only if' in equation (27) based on a finite net of data, and the quantitative equivalence ${Q}_{A}^{{M}_{1}}({\rho }_{{AB}})={Q}_{A}^{{G}_{1}}({\rho }_{{AB}})$ was demonstrated for two qubits. The experiment in [193] started instead with a classically correlated state, in which QCs were then created by local noise (using dissipative non-LCPO maps [64]), and finally activated into entanglement with ancillae through the protocol described in section 2.3.2. Both experiments measured QCs via the negativity of quantumness. Finally, the negativity of quantumness has been studied for Gaussian states in [121], where it was also shown that non-Gaussian pre-measurement interactions are necessary for the entanglement activation of QCs in Gaussian states. Alternative schemes inspired by the activation protocol have been proposed for continuous variable systems [194, 195].

The unitary response measure of QCs ${Q}_{A}^{{U}_{2}}$ was then defined by resorting to the squared Hilbert–Schmidt distance in [39, 40] and was shown to be equivalent to the Hilbert–Schmidt geometric measure of QCs ${Q}_{A}^{{G}_{2}}$ when subsystem A is a qubit, as recalled in equation (68). Consequently, due to the problem of equation (69) highlighted in [111], the response measure ${Q}_{A}^{{U}_{2}}$ does not satisfy requirement (iv) and does not qualify as a valid measure of QCs.

The trace distance response measure of QCs ${Q}_{A}^{{U}_{1}}$ was studied in [80] and connected both to the trace distance geometric measure of QCs and to the negativity of quantumness defined via the entanglement activation approach, when the subsystem A is a qubit, see sections 3.3.3 and 3.6.2.

In [41] the unitary response measures ${Q}_{A}^{{U}_{\delta }}$, therein referred to as 'discord of response', were defined more generally as in equation (59) by resorting to any contractive distance D_δ, with a particular focus on the squared Bures distance (see table 2). Such a study was then complemented in [56] with a comprehensive comparison between several distance-based measures of QCs, including notable advances for the evaluation of the unitary response measures ${Q}_{A}^{{U}_{\mathrm{Bu}}}$ and ${Q}_{A}^{{U}_{\mathrm{He}}}$ based respectively on the squared Bures and the squared Hellinger distances (see table 2). In particular, for pure states ${\left|| ,\psi ,\rangle \right.\rangle }_{{AB}}$, it holds [56]

$$\begin{eqnarray}&&{Q}_{A}^{{U}_{\mathrm{Bu}}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})=2\left((,1-\sqrt{1-{E}^{{U}_{F}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})},)\right),\end{eqnarray} \tag{ 116 }$$

$$\begin{eqnarray}&&{Q}_{A}^{{U}_{\mathrm{He}}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})=2{E}^{{U}_{F}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}}),\end{eqnarray} \tag{ 117 }$$

where

$$\begin{eqnarray}&&{E}^{{U}_{F}}({\left|| ,\psi ,\rangle \right.\rangle }_{{AB}})=\mathop{\inf }\limits_{{U}_{A}}\left[[,1-{\left|| ,\left.\langle \langle ,\psi ,(\right({U}_{A}\otimes {{\mathbb{I}}}_{B}){\left|| ,\psi ,\rangle \right.\rangle }_{{AB}},| \right|}^{2},]\right]\end{eqnarray} \tag{ 118 }$$

is the entanglement of response defined as in equation (60) with the infidelity (see table 2) chosen as distance function and with optimisation restricted to root-of-unity local unitaries U_A. In the case of mixed states, closed formulae for ${Q}_{A}^{{U}_{\mathrm{Bu}}}$ and ${Q}_{A}^{{U}_{\mathrm{He}}}$ were first derived in [27, 41] for the special case of two-qubit Bell diagonal states. More generally, for arbitrary states ${\rho }_{{AB}}$ when A is a qubit, these two measures can be linked analytically to their geometric counterparts by the simple relation [56]

$$\begin{eqnarray}&&{Q}_{A}^{{U}_{\delta }}({\rho }_{{AB}})=4{Q}_{A}^{{G}_{\delta }}({\rho }_{{AB}})-{\left[[,{Q}_{A}^{{G}_{\delta }}({\rho }_{{AB}}),]\right]}^{2},\end{eqnarray} \tag{ 119 }$$

with $\delta =\mathrm{Bu},\mathrm{He}$. In particular, by chaining this with equations (75) and (67), we get the simple relation

$$\begin{eqnarray}&&{Q}_{A}^{{U}_{\mathrm{He}}}({\rho }_{{AB}})=4{Q}_{A}^{{G}_{2}}(\sqrt{{\rho }_{{AB}}})=4{Q}_{A}^{{M}_{2}}(\sqrt{{\rho }_{{AB}}}),\end{eqnarray} \tag{ 120 }$$

which shows that the unitary response measure of QCs based on the squared Hellinger distance is equivalent to the remedied Hilbert–Schmidt geometric measure defined in [113], hence is analytically computable for all states ${\rho }_{{AB}}$ when A is a qubit. In this specific case, the Hellinger unitary response measure ${Q}_{A}^{{U}_{\mathrm{He}}}({\rho }_{{AB}})$ is also equivalent to the coherence based measure defined in terms of the Wigner–Yanase skew information ${Q}_{A}^{{C}^{\mathrm{WY}}}({\rho }_{{AB}})$, introduced in [54] as 'local quantum uncertainty' (see section 3.8.4). Namely,

$$\begin{eqnarray}&&{Q}_{A}^{{U}_{\mathrm{He}}}({\rho }_{{AB}})=2{Q}_{A}^{{C}^{\mathrm{WY}}}({\rho }_{{AB}}),\end{eqnarray} \tag{ 121 }$$

for all states ${\rho }_{{AB}}$ when A is a qubit. In this case, such an observation independently provides a closed formula for ${Q}_{A}^{{U}_{\mathrm{He}}}({\rho }_{{AB}})$ [54], equivalent to the one given in [56, 113].

The Bures and Hellinger unitary response measures have been studied for Gaussian states with the restriction to Gaussian unitaries in [197], within the operational context of quantum reading (see section 4.2.2).

Finally, in [58] the restriction to a harmonic spectrum was lifted by defining the 'discriminating strength' as follows

$$\begin{eqnarray}&&{Q}_{A}^{{U}_{{\rm{C}}}^{{\rm{\Gamma }}}}({\rho }_{{AB}}):= 1-\mathop{\max }\limits_{{U}_{A}^{{\rm{\Gamma }}}}{\rm{C}}\left((,{\rho }_{{AB}},{U}_{A}^{{\rm{\Gamma }}}[{\rho }_{{AB}}],)\right),\end{eqnarray} \tag{ 122 }$$

where

$$\begin{eqnarray}&&{\rm{C}}(\rho ,\sigma ):= \mathop{\min }\limits_{0\leqslant s\leqslant 1}{\rm{Tr}}({\rho }^{s}{\sigma }^{1-s})\end{eqnarray} \tag{ 123 }$$

is a measure of indistinguishability between the states ρ and σ, which coincides with the Uhlmann fidelity F [198] (see table 2),

$$\begin{eqnarray}&&F(\rho ,\sigma ):= {\left[[,{\rm{Tr}}\left((,\sqrt{\sqrt{\sigma }\rho \sqrt{\sigma }},)\right),]\right]}^{2},\end{eqnarray} \tag{ 124 }$$

when either ρ or σ is pure. The quantity in equation (123) enters in the definition of the quantum Chernoff bound [132]

$$\begin{eqnarray}&&\xi (\rho ,\sigma ):= -\mathrm{log}\,{\rm{C}}(\rho ,\sigma ).\end{eqnarray} \tag{ 125 }$$

In equation (122), the optimisation is over all local unitaries ${U}_{A}^{{\rm{\Gamma }}}$ on subsystem A with non-degenerate spectrum Γ, as in equation (58). The discriminating strength, whose defining operational interpretation in the context of channel discrimination will be discussed in section 4.2.2, can be thus regarded as a unitary response measure of QCs based on the Chernoff distance ${D}_{{\rm{C}}}(\rho ,\sigma ):= 1-{\rm{C}}(\rho ,\sigma )$. The choice of the spectrum Γ in order to maximise the measure ${Q}_{A}^{{U}_{{\rm{C}}}^{{\rm{\Gamma }}}}$ was analysed as well in [58]. When subsystem A is a qubit (in which case any spectrum leads to an equivalent measure up to a normalisation factor, hence the harmonic spectrum can be assumed without any loss of generality [54]), it was further shown in [58] that the discriminating strength coincides up to a constant multiplicative factor with the local quantum uncertainty [54], and thus with the Hellinger unitary response measure of QCs, via equation (121). In continuous variable systems, the discriminating strength was investigated in [199] by restricting to a minimisation over Gaussian unitaries, and a closed expression was derived for a subclass of two-mode Gaussian states, exploiting the analytical formula for the quantum Chernoff distance between a pair of two-mode Gaussian states obtained in [200].

At present, the status of requirement (v) for all the above mentioned unitary response measures of QCs remains unknown.

As with many other resources in quantum information theory, the relative entropy can be used to define a coherence quantifier ${C}^{\mathrm{RE}}$ which is a full monotone in any possible resource theory of coherence [42, 43], and is easily computable for all quantum states as indicated in table 5. Following the approach outlined in section 3.2.6, we can define basis-dependent quantifiers of local coherence with respect to the relative entropy by adopting a geometric approach [47, 95], and simplify their expression by resorting to the equivalence with the measurement induced geometric approach [93]. We have then

$$\begin{eqnarray}{C}_{A}^{\mathrm{RE}\ \{{\left|| ,a,\rangle \right.\rangle }_{A}\}}({\rho }_{{AB}}) & := & \mathop{\inf }\limits_{{\chi }_{{AB}}\in {I}_{A}^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}}{D}_{\mathrm{RE}}({\rho }_{{AB}},{\chi }_{{AB}})={D}_{\mathrm{RE}}\left((,{\rho }_{{AB}},{\pi }_{A}[{\rho }_{{AB}}],)\right),\\ {C}_{{AB}}^{\mathrm{RE}\ \{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}}({\rho }_{{AB}}) & := & \mathop{\inf }\limits_{{\chi }_{{AB}}\in {I}_{{AB}}^{\{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}}}{D}_{\mathrm{RE}}({\rho }_{{AB}},{\chi }_{{AB}})={D}_{\mathrm{RE}}\left((,{\rho }_{{AB}},{\pi }_{{AB}}[{\rho }_{{AB}}],)\right),\end{eqnarray} \tag{ 126 }$$

where, as usual, the LPMs ${\pi }_{A}$ and ${\pi }_{{AB}}$ project the subsystems in the chosen reference bases $\{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}$ as in equations (18) and (19). The coherence based measures of QCs defined in terms of relative entropy are then obtained upon minimisation over said local bases as in equation (61), and as anticipated return the relative entropy based geometric measures of QCs,

$$\begin{eqnarray}{Q}_{A}^{{C}^{\mathrm{RE}}}({\rho }_{{AB}}) & = & \mathop{\inf }\limits_{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}{C}_{A}^{\mathrm{RE}\ \{{\left|| ,a,\rangle \right.\rangle }_{A}\}}({\rho }_{{AB}})={Q}_{A}^{{G}_{\mathrm{RE}}}({\rho }_{{AB}}),\\ {Q}_{{AB}}^{{C}^{\mathrm{RE}}}({\rho }_{{AB}}) & = & \mathop{\inf }\limits_{\{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}}{C}_{{AB}}^{\mathrm{RE}\ \{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}}({\rho }_{{AB}})={Q}_{{AB}}^{{G}_{\mathrm{RE}}}({\rho }_{{AB}}),\end{eqnarray} \tag{ 127 }$$

discussed in section 3.3.1 together with all their equivalent interpretations in terms of the other types of QCs quantifiers.

Exploiting the results of [35, 49], one can also derive coherence based measures of QCs in terms of infidelity (see table 5), which turn out to match their geometric counterparts, analogously to the case of relative entropy. Compactly, we have indeed

$$\begin{eqnarray}{Q}_{A}^{{C}^{F}}({\rho }_{{AB}}) & := & 1-\mathop{\sup }\limits_{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}\ \mathop{\sup }\limits_{{\chi }_{{AB}}\in {I}_{A}^{\{{\left|| ,a,\rangle \right.\rangle }_{A}\}}}F({\rho }_{{AB}},{\chi }_{{AB}})={Q}_{A}^{{G}_{F}}({\rho }_{{AB}}),\\ {Q}_{{AB}}^{{C}^{F}}({\rho }_{{AB}}) & := & 1-\mathop{\sup }\limits_{\{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}}\ \mathop{\sup }\limits_{{\chi }_{{AB}}\in {I}_{{AB}}^{\{{\left|| ,a,\rangle \right.\rangle }_{A},{\left|| ,b,\rangle \right.\rangle }_{B}\}}}{D}_{F}({\rho }_{{AB}},{\chi }_{{AB}})={Q}_{{AB}}^{{G}_{F}}({\rho }_{{AB}}),\end{eqnarray} \tag{ 128 }$$

where the infidelity based geometric measures are monotonically related to the corresponding measures defined in terms of squared Bures distance (see table 2) and enjoy accordingly all their properties, discussed in section 3.3.4.

The ℓ₁ norm defines a very intuitive and easy to compute quantifier of coherence (see table 5), which is also a full monotone in all possible resource theories of quantum coherence [43]. A consistent pathway to obtain QCs measures from the ℓ₁ norm of coherence has been presented in section 3.6.2, where the ensuing coherence based quantifiers have been shown to reproduce exactly the entanglement activation measures based on negativity. In particular, the two-sided measure ${Q}_{{AB}}^{{C}^{{{\ell }}_{1}}}({\rho }_{{AB}})$, alias the two-sided negativity of quantumness ${Q}_{{AB}}^{{E}^{N}}({\rho }_{{AB}})$ [80], exhibits a particularly appealing form, as it amounts to the total of the moduli of the off-diagonal elements of the density matrix ${\rho }_{{AB}}$ (traditionally referred to as 'coherences'), minimised over all local product bases. This is valid for arbitrarily large multipartite systems as well, with respect to the product of orthonormal bases on each subsystem [36].

Beyond redefinition of existing measures, the coherence based approach can be used to define original measures of QCs of high operational significance, which do not follow straightforwardly from other approaches. This is the case in particular for two coherence quantifiers defined in the context of the resource theory of asymmetry, namely the Wigner–Yanase skew information and the quantum Fisher information (see table 5). Both quantities are monotones under phase covariant operations [69], but not under more general incoherent operations [43], as proven in [82, 83] for the skew information, and in [70] for the quantum Fisher information. These two metrics belong to the family of monotone and Riemannian metrics discussed in section 3.3.7 with geodesics corresponding respectively to the Hellinger and the Bures distances (see table 2) [77]. In [54] and [57], valid measures of QCs have been successfully defined based on these two asymmetry monotones, as we now review within a unified approach.

Given a measure of coherence in the context of quantum asymmetry ${C}^{\eta {\rm{\Gamma }}}(\rho ,{K}^{{\rm{\Gamma }}})$, defined with respect to a fixed reference basis $\{\left|| ,i,\rangle \right.\rangle \}$ identified as the eigenbasis of the observable ${K}^{{\rm{\Gamma }}}={\sum }_{i}{k}_{i}\left|| ,i,\rangle \right.\rangle \,\left.\langle \langle ,i,| \right|$ with non-degenerate spectrum ${\rm{\Gamma }}=\{{k}_{i}\}$, then the corresponding one-sided measure of QCs of a bipartite state ${\rho }_{{AB}}$ can be defined adapting equation (61) as

$$\begin{eqnarray}&&{Q}_{A}^{{C}^{\eta }\ {\rm{\Gamma }}}({\rho }_{{AB}}):= \mathop{\inf }\limits_{{K}_{A}^{{\rm{\Gamma }}}}{C}^{\eta {\rm{\Gamma }}}({\rho }_{{AB}},{K}_{A}^{{\rm{\Gamma }}}\otimes {{\mathbb{I}}}_{B}),\end{eqnarray} \tag{ 129 }$$

where the minimisation is over all local observables ${K}_{A}^{{\rm{\Gamma }}}$ on subsystem A with non-degenerate spectrum Γ. Notice the resemblance with the case of unitary response measures in equation (58). Similarly to such a case, in fact, it is not immediate to extend equation (129) to define a corresponding two-sided measure of QCs, as including an optimisation over observables ${K}_{B}^{{\rm{\Gamma }}}$ on B as well might nullify the resulting two-sided quantifier even on maximally entangled states, like in the instance of equation (32).

Using equation (129), we can now define the 'local quantum uncertainty' ${Q}_{A}^{{C}^{\mathrm{WY}}\ {\rm{\Gamma }}}({\rho }_{{AB}})$ as the coherence based measure of QCs induced by the Wigner–Yanase skew information [54], and the 'interferometric power' ${Q}_{A}^{{C}^{\mathrm{QF}}\ {\rm{\Gamma }}}({\rho }_{{AB}})$ as the coherence based measure of QCs induced by the quantum Fisher information [57]. More precisely, each of these expressions defines a family of measures, dependent upon the fixed spectrum Γ of the observables entering the definition. The original names derive from their physical meaning. As the skew information ${C}^{\mathrm{WY}}(\rho ,K)$ can be regarded as a quantifier of the genuinely quantum share of the uncertainty (out of the total variance) associated to the measurement of K on the state ρ, which is due to the fact that $[\rho ,K]\ne 0$ [201], the QCs measure ${Q}_{A}^{{C}^{\mathrm{WY}}}$ consequently quantifies the minimum quantum uncertainty affecting the measurement of a local observable on subsystem A in the state ${\rho }_{{AB}};$ in this way, QCs are clearly linked to non-commutativity, as classical–quantum states are all and only the states which can commute with a local observable on A [54]. On the other hand, the interferometric power ${Q}_{A}^{{C}^{\mathrm{QF}}\ {\rm{\Gamma }}}({\rho }_{{AB}})$ recognises this non-commutativity as an advantageous feature to maintain a guaranteed precision in a metrological task, as we will detail in section 4.2.1.

Beyond the specific interpretations, however, both measures enjoy similar properties, due to their common origin. In particular, they both obey all requirements (i)–(iv) [54, 57], while requirement (v) is currently being tested, with some partial progress achieved for ${Q}_{A}^{{C}^{\mathrm{QF}}\ {\rm{\Gamma }}}({\rho }_{{AB}})$ [67]. By construction, it holds

$$\begin{eqnarray}&&{Q}_{A}^{{C}^{\mathrm{WY}}\ {\rm{\Gamma }}}({\rho }_{{AB}})\leqslant \displaystyle \frac{1}{4}{Q}_{A}^{{C}^{\mathrm{QF}}\ {\rm{\Gamma }}}({\rho }_{{AB}}),\end{eqnarray} \tag{ 130 }$$

with equality on pure states (for this reason, the $\tfrac{1}{4}$ factor is usually included in the definition of the interferometric power for normalisation) [57, 202]. Closed analytical formulae are available for both measures on all bipartite states ${\rho }_{{AB}}$ when subsystem A is a qubit [54, 57], demonstrating that these measures reconcile the typically contrasting requirements of computability and reliability [96]. In this case (dropping the spectrum superscript without loss of generality as discussed in section 3.2.5), the local quantum uncertainty ${Q}_{A}^{{C}^{\mathrm{WY}}}({\rho }_{{AB}})$ is also equivalent to the unitary response measure based on the Hellinger distance ${Q}_{A}^{{U}_{\mathrm{He}}}({\rho }_{{AB}})$ [54] (which in turn reduces to the remedied Hilbert–Schmidt geometric measure ${Q}_{A}^{{G}_{2}}(\sqrt{{\rho }_{{AB}}})$ [113]), and to the discriminating strength ${Q}_{A}^{{U}_{{\rm{C}}}^{{\rm{\Gamma }}}}({\rho }_{{AB}})$ [58], as discussed in section 3.7. Surprisingly, a similar equivalence does not hold between the interferometric power ${Q}_{A}^{{C}^{\mathrm{QF}}}({\rho }_{{AB}})$ and the unitary response measure based on the Bures distance, and it is still unknown whether the former may admit any alternative interpretation in terms of a geometric or a unitary response type measure. The interferometric power has been extended to continuous variable systems, and a closed formula derived for all two-mode Gaussian states restricting to an optimisation over Gaussian observables K with harmonic spectrum [202].

Finally, the interferometric power was measured experimentally in a two-qubit nuclear magnetic resonance ensemble realised in chloroform, as part of an implementation of black box quantum phase estimation with noisy resources (see 4.2.1) [57].

Given that the formalism of coherence based measures of QCs has been introduced only in the present paper in its full generality, there have been no further records (to the best of our knowledge) to define other quantifiers of this type. However, the procedure outlined in section 3.2.6 could be applied to any valid coherence measure in principle, with the analysis in section 3.8.4 being specifically useful for monotones arising from the resource theory of asymmetry specialised to coherence. In this context, let us mention that the authors of [203] defined the 'quantum variance', a related quantifier of quantum uncertainty yielding a computable and experimentally accessible lower bound to both Wigner–Yanase skew information and quantum Fisher information, and discussed its applications to measuring QCs and to exploring the associated quantum coherence length in many-body systems.

One suggestion for the future could be to define and study the measures of QCs based on the recently introduced 'robustness of coherence' [45] (see table 5), which may be expected to enjoy some appealing operational interpretations in a channel discrimination setting. This is left open for a keen reader (or co-author). Another possibility could be to study the measures of QCs induced by Tsallis generalisations of the relative entropy of coherence [204], and investigate their relationship with corresponding geometric and informational measures defined via the same entropies.

The 'surprisal of measurement recoverability' ${Q}_{A}^{{R}_{F}}$ has been originally defined by resorting to the (negative logarithm of) Uhlmann fidelity [50], corresponding the one-sided definition in equations (62) with ${D}_{\delta }(\rho ,\sigma )=-\mathrm{log}F(\rho ,\sigma )$, where the fidelity $F(\rho ,\sigma )$ is given by equation (124). Explicitly,

$$\begin{eqnarray}&&{Q}_{A}^{{R}_{F}}({\rho }_{{AB}}):= -\mathrm{log}\mathop{\sup }\limits_{{{\rm{\Lambda }}}_{A}^{\mathrm{EB}}}F\left((,{\rho }_{{AB}},{{\rm{\Lambda }}}_{A}^{\mathrm{EB}}[{\rho }_{{AB}}],)\right),\end{eqnarray} \tag{ 131 }$$

where the optimisation is over the convex set of entanglement-breaking channels ${{\rm{\Lambda }}}_{A}^{\mathrm{EB}}$ acting on subsystem A

The motivation behind this definition arose from an alternative angle from which the quantum discord (see section 3.5.2) can be looked at, introduced for the first time in [111]. Indeed, by using the fact that any LGM ${\tilde{{\rm{\Pi }}}}_{A}$ on subsystem A can be written as a unitary ${U}_{A\to A^{\prime} C}^{{\tilde{{\rm{\Pi }}}}_{A}}$ from A to a composite system $A^{\prime} C$ followed by discarding C, i.e., ${\tilde{{\rm{\Pi }}}}_{A}[{\rho }_{{AB}}]={{\rm{Tr}}}_{C}\left((,{U}_{A\to A^{\prime} C}^{{\tilde{{\rm{\Pi }}}}_{A}}[{\rho }_{{AB}}],)\right)$, one then gets

$$\begin{eqnarray}&&{Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}})=\mathop{\inf }\limits_{{\tilde{{\rm{\Pi }}}}_{A}}{{ \mathcal I }}_{{A}^{\prime }}\left((,{U}_{A\to A^{\prime} C}^{{\tilde{{\rm{\Pi }}}}_{A}}[{\rho }_{{AB}}],)\right),\end{eqnarray} \tag{ 132 }$$

where we denote by ${{ \mathcal I }}_{A}({\rho }_{{ABC}})$ the conditional quantum mutual information of the tripartite state ${\rho }_{{ABC}}$ with respect to subsystem A, defined as

$$\begin{eqnarray}&&{{ \mathcal I }}_{A}({\rho }_{{ABC}}):= { \mathcal S }({\rho }_{{AB}})+{ \mathcal S }({\rho }_{{AC}})-{ \mathcal S }({\rho }_{A})-{ \mathcal S }({\rho }_{{ABC}}).\end{eqnarray} \tag{ 133 }$$

The latter quantity, often denoted as ${ \mathcal I }{(B;C| A)}_{\rho }$ in information theory literature [205], captures the correlations present between B and C from the perspective of A in the i.i.d. resource limit, where an asymptotically large number of copies of the tripartite state ${\rho }_{{ABC}}$ are shared between the three parties.

In the quest for developing a version of the conditional quantum mutual information which could be relevant for the finite resource regime, in [50] the surprisal of the fidelity of recovery of a tripartite quantum state ${\rho }_{{ABC}}$ with respect to subsystem A was defined as follows,

$$\begin{eqnarray}&&{F}_{A}^{R}({\rho }_{{ABC}}):= -\mathrm{log}\mathop{\sup }\limits_{{{\rm{\Lambda }}}_{A\to {AC}}^{R}}F\left((,{\rho }_{{ABC}},{{\rm{\Lambda }}}_{A\to {AC}}^{R}[{\rho }_{{ABC}}],)\right),\end{eqnarray} \tag{ 134 }$$

where the optimisation is over all recovery maps ${{\rm{\Lambda }}}_{A\to {AC}}^{R}$ from A to the composite system AC. The surprisal of measurement recoverability was then introduced in [50] in complete analogy with quantum discord by simply substituting the conditional quantum mutual information with the surprisal of the fidelity of recovery,

$$\begin{eqnarray}&&{Q}_{A}^{{R}_{F}}({\rho }_{{AB}}):= \mathop{\inf }\limits_{{\tilde{{\rm{\Pi }}}}_{A}}{F}_{{A}^{\prime }}^{R}\left((,{U}_{A\to A^{\prime} C}^{{\tilde{{\rm{\Pi }}}}_{A}}[{\rho }_{{AB}}],)\right),\end{eqnarray} \tag{ 135 }$$

and only afterwards proven to be equivalent to equation (131).

For bipartite pure states ${\left|| ,\psi ,\rangle \right.\rangle }_{{AB}}$, the surprisal of measurement recoverability is equal to an entanglement measure,

$$\begin{eqnarray}&&{Q}_{A}^{{R}_{F}}(| {\psi }_{{AB}}\rangle )=-\mathrm{log}{\rm{Tr}}({\rho }_{A}^{2}),\end{eqnarray} \tag{ 136 }$$

where ${\rho }_{A}$ is the marginal state of subsystem A, thus satisfying requirement (iii). The validity of requirement (v) is still unknown.

In general, it was shown in [50] using the results of [206] that the surprisal of measurement recoverability is a lower bound to the quantum discord,

$$\begin{eqnarray}&&{Q}_{A}^{{R}_{F}}({\rho }_{{AB}})\leqslant {Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}}).\end{eqnarray} \tag{ 137 }$$

Furthermore, [207] developed a hierarchy of efficiently computable and faithful lower bounds to ${Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}$ that converge exactly to ${Q}_{A}^{{R}_{F}}$, thus showing in particular that the surprisal of measurement recoverability can be bounded numerically by a semidefinite programme, and admits an operational interpretation in the task of local broadcasting, as we detail in section 4.1.1.

It will be interesting in the future to study two-sided versions ${Q}_{{AB}}^{{R}_{F}}({\rho }_{{AB}})$ of the surprisal of measurement recoverability, as well as other recoverability measures corresponding to different choices of distance D_δ in equations (62), and explore their interplay with the other types of measures of QCs collected in this review.

Further related measures can be obtained by alternative generalisations of the conditional quantum mutual information, e.g. in terms of Rényi entropies [208].

Copying information is a natural ability in our classical realm. Indeed, the classical information you are reading right now is one of (hopefully) many copies. In the quantum realm, however, unconditional copying is not allowed, due to the no-cloning theorem [211, 212]. The no-cloning theorem says that it is impossible to create an identical copy of an arbitrary unknown pure quantum state of a system by using a composite unitary acting upon the system and an ancilla. The best that we can do is to reliably clone orthogonal pure states, which effectively corresponds to copying of classical information. Such an impossibility was subsequently generalised to mixed states and general transformations leading to the so called no-broadcasting theorem [213].

These no-go theorems enshrine classicality in single systems. For composite systems, the task of 'local broadcasting' was outlined in [24]. Consider a composite bipartite state ${\rho }_{{AB}}$ shared between laboratory A and B, with each laboratory also possessing an ancilla, respectively $A^{\prime}$ and $B^{\prime}$, in the composite state ${\sigma }_{A^{\prime} B^{\prime} }=\left|| ,0,\rangle \right.\rangle {\left.\langle \langle ,0,| \right|}_{{A}^{\prime }}\otimes \left|| ,0,\rangle \right.\rangle {\left.\langle \langle ,0,| \right|}_{{B}^{\prime }}$. The objective is for A and B to perform local operations only (and no communication) on the joint state ${\rho }_{{AB}}\otimes {\sigma }_{A^{\prime} B^{\prime} }$, in order to produce a joint state ${\rho }_{{AA}^{\prime} {BB}^{\prime} }=({{\rm{\Lambda }}}_{{AA}^{\prime} }\otimes {{\rm{\Lambda }}}_{{BB}^{\prime} })[{\rho }_{{AB}}\otimes {\sigma }_{A^{\prime} B^{\prime} }]$ which contains two (possibly correlated) copies of ${\rho }_{{AB}}$,

$$\begin{eqnarray}&&{{\rm{Tr}}}_{A^{\prime} B^{\prime} }({\rho }_{{AA}^{\prime} {BB}^{\prime} })={{\rm{Tr}}}_{{AB}}({\rho }_{{AA}^{\prime} {BB}^{\prime} })={\rho }_{{AB}}.\end{eqnarray} \tag{ 138 }$$

It was shown in [24] that this task can only be performed if ${\rho }_{{AB}}\in {C}_{{AB}}$, which is a very intuitive result saying that only classically correlated states can be locally broadcast. More generally, it was shown that, even if the task was relaxed to that of obtaining two (possibly different) reduced states but with the same total correlations as measured by the mutual information between A and B (or $A^{\prime}$ and $B^{\prime}$), then the correlations themselves could only be broadcast if ${\rho }_{{AB}}$ is classically correlated,

$$\begin{eqnarray*}&&{\rho }_{{AB}}\in {C}_{{AB}}\,\,\leftrightarrow \,\,\,\exists \,{\rho }_{{AA}^{\prime} {BB}^{\prime} }\,\,\,{\rm{such}}\,{\rm{that}}\,\,\,{ \mathcal I }\left((,{{\rm{Tr}}}_{A^{\prime} B^{\prime} }({\rho }_{{AA}^{\prime} {BB}^{\prime} }),)\right)={ \mathcal I }\left((,{{\rm{Tr}}}_{{AB}}({\rho }_{{AA}^{\prime} {BB}^{\prime} }),)\right)={ \mathcal I }\left((,{\rho }_{{AB}},)\right),\end{eqnarray*}$$

which is called the no-local-broadcasting theorem [24]. One-sided local (or unilocal) broadcasting was then considered in [214]. Here, the idea is for only A to have access to an ancilla $\left|| ,0,\rangle \right.\rangle {\left.\langle \langle ,0,| \right|}_{{A}^{\prime }}$ and to perform local operations to obtain states ${\rho }_{{AA}^{\prime} B}=({{\rm{\Lambda }}}_{{AA}^{\prime} }\otimes {{\mathbb{I}}}_{B})[{\rho }_{{AB}}\otimes \left|| ,0,\rangle \right.\rangle {\left.\langle \langle ,0,| \right|}_{{A}^{\prime }}]$ such that

$$\begin{eqnarray}&&{ \mathcal I }\left((,{{\rm{Tr}}}_{{A}^{\prime }}({\rho }_{{AA}^{\prime} B}),)\right)={ \mathcal I }\left((,{{\rm{Tr}}}_{A}({\rho }_{{AA}^{\prime} B}),)\right)={ \mathcal I }\left((,{\rho }_{{AB}},)\right).\end{eqnarray} \tag{ 139 }$$

As intuitively expected, this is shown to be possible if and only if ${\rho }_{{AB}}\in {C}_{A}$. Partial broadcasting of QCs was further discussed in [215].

By design it appears straight away that QCs are relevant in the local broadcasting paradigm: indeed, the (im)possibility of local or unilocal broadcasting provides another qualitative characterisation of classical versus quantum states, that could be added to figure 3. Can such a paradigm provide natural operational interpretations to measures of QCs then? The answer is affirmative.

To begin with, precise quantitative links to one-sided informational measures of QCs and unilocal broadcasting were provided recently by [216], who consider a bipartite system in the state ${\rho }_{{AB}}$ and the process of party A redistributing their correlations with B to N ancillae ${\{{A}_{i}\}}_{i=1}^{N}$ by local operations ${{\rm{\Lambda }}}_{A\to {A}_{1}\ldots {A}_{N}}$, with the aim to maintain as many correlations as possible between each A_i and the other party B. Let us define the reduced state of A_i and B after the redistribution as

$$\begin{eqnarray}&&{\rho }_{{A}_{i}B}={{\rm{Tr}}}_{{\otimes }_{j\ne i}{A}_{j}}\left[[,({{\rm{\Lambda }}}_{A\to {A}_{1}\ldots {A}_{N}}\otimes {{\mathbb{I}}}_{B})[{\rho }_{{AB}}],]\right].\end{eqnarray} \tag{ 140 }$$

Due the no-unilocal broadcasting theorem [24, 214], for any ${\rho }_{{AB}}\,\not\hspace{-2pt}{\in }\,{C}_{A}$ we know that some correlations will be generally lost in the redistribution process, i.e., each part A_i will have less correlations with B than the initial correlations between A and B, or in formulae, ${ \mathcal I }({\rho }_{{A}_{i}B})\leqslant { \mathcal I }({\rho }_{{AB}})$, $\forall \ i=1,\ldots ,N$. The loss of correlations when redistributing to the ancilla A_i can be averaged over all the ancillae, and minimised over all the local redistributing operations, to arrive at the minimum average loss of correlations,

$$\begin{eqnarray}&&{\overline{{\rm{\Delta }}}}_{A}^{(N)}({\rho }_{{AB}}):= \mathop{\min }\limits_{{{\rm{\Lambda }}}_{A\to {A}_{1}\ldots {A}_{N}}}\displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{N}\left[[,{ \mathcal I }({\rho }_{{AB}})-{ \mathcal I }({\rho }_{{A}_{i}B}),]\right].\end{eqnarray} \tag{ 141 }$$

Quite remarkably, in the limit of infinitely many ancillae, it was shown that [216]

$$\begin{eqnarray}&&\mathop{\mathrm{lim}}\limits_{N\to \infty }{\overline{{\rm{\Delta }}}}_{A}^{(N)}({\rho }_{{AB}})={Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}}),\end{eqnarray} \tag{ 142 }$$

that is, the asymptotic minimum average loss of correlations after attempting to unilocally broadcast any quantum state ${\rho }_{{AB}}$ is given by its QCs content as measured by the quantum discord. Equation (142) yields one of the most prominent operational interpretations of the quantum discord, and generalises the similar work of [217] whose result was limited to pure states only. Borrowing the title of [217], this means that quantum discord cannot be shared, while only the classical correlations quantified by equation (90) can be arbitrarily redistributed to many parties on one side.

These results have intimate connections to 'quantum Darwinism' [218]. The idea behind quantum Darwinism is to explain the emergence of classical reality by observing a quantum system indirectly through its environment. The only information objectively accessible about the quantum system is that proliferated into many sub-parts of the environment. Thus, it is clear that ${\overline{{\rm{\Delta }}}}_{A}^{(N)}({\rho }_{{AB}})$, hence the quantum discord of ${\rho }_{{AB}}$ in the limit of asymptotically many sub-parts of the environment, represents the portion of correlations inaccessible to the classical realm [216].

Curiously, no rigorous result analogous to equation (142) has been yet established in the two-sided setting of redistributing correlations in a quantum state ${\rho }_{{AB}}$ both on $A\to {\{{A}_{i}\}}_{i=1}^{N}$ and on $B\to {\{{B}_{j}\}}_{j=1}^{N}$. It is reasonable to expect that in this case one would recover the two-sided informational measure ${Q}_{{AB}}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}})$ as the asymptotic minimum average of the correlation gap ${ \mathcal I }({\rho }_{{AB}})-{ \mathcal I }({\rho }_{{A}_{i}{B}_{i}})$. We conjecture this to be the case, following the intuition that ${Q}_{{AB}}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}})$ should be related to the amount of correlations lost when performing local broadcasting [24], since the maximum share of correlations in a state ${\rho }_{{AB}}$ that can be broadcast, i.e., transferred to classical registers from both subsystems, is given by the classical mutual information of ${\rho }_{{AB}}$, equation (94). We hope our conjecture will be settled in the near future.

Another way to obtain quantitative results is to look at how similar a locally broadcast copy can be to the original state ${\rho }_{{AB}}$ [207]. Focusing again on the unilocal broadcasting setting of A attempting to locally copy ${\rho }_{{AB}}$ to N ancillae ${\{{A}_{i}\}}_{i=1}^{N}$ and arriving at the reduced states ${\rho }_{{A}_{i}B}$ in equation (140), it is clear that for any A_i

$$\begin{eqnarray}&&\mathop{\max }\limits_{{{\rm{\Lambda }}}_{A\to {A}_{1}\ldots {A}_{N}}}F({\rho }_{{AB}},{\rho }_{{A}_{i}B})\leqslant 1,\end{eqnarray} \tag{ 143 }$$

where F is the Uhlmann fidelity, equation (124). Equality holds if and only if ${\rho }_{{AB}}$ can be locally broadcast, or equivalently if ${\rho }_{{AB}}\in {C}_{A}$: in this case, for ${\rho }_{{AB}}={\sum }_{i}{p}_{i}\left|| ,i,\rangle \right.\rangle {\left.\langle \langle ,i,| \right|}_{A}\otimes {\rho }_{B}^{(i)}$, a channel implementing the transformation ${\left|| ,i,\rangle \right.\rangle }_{A}\to {\left|| ,i,\rangle \right.\rangle }_{A}^{\otimes N}$ would be suitable for broadcasting. This transformation is a type of N-Bose-symmetric channel ${{\rm{\Lambda }}}_{A\to {A}_{1}\ldots {A}_{N}}^{{B}_{N}}$ mapping A onto the fully symmetric subspace of ${A}_{1}\ldots {A}_{N}$, i.e. returning a density operator invariant under any permutation $\varpi ({A}_{1}\ldots {A}_{N})$ of the ancillae [219]. It is then interesting to consider [207]

$$\begin{eqnarray}&&{Q}_{A}^{{{\rm{B}}}_{N}}({\rho }_{{AB}}):= -\mathrm{log}\mathop{\sup }\limits_{{{\rm{\Lambda }}}_{A\to {A}_{1}\ldots {A}_{N}}^{{B}_{N}}}F({\rho }_{{AB}},{\rho }_{{A}_{1}B})\end{eqnarray} \tag{ 144 }$$

as a figure of merit quantifying the quality of N-copy local broacasting (note that, due to the symmetry of ${{\rm{\Lambda }}}_{A\to {A}_{1}\ldots {A}_{N}}^{{B}_{N}}$, it is enough to just consider ${\rho }_{{A}_{1}B}$). In the case $N\to \infty$, the composition of ${{\rm{\Lambda }}}_{A\to {A}_{1}\ldots {A}_{N}}^{{B}_{N}}$ followed by the partial trace over all but ancilla A₁ is equivalent to the action of an entanglement breaking channel, equation (39), or simply ${\rho }_{{A}_{1}B}$ is the output of an entanglement breaking channel acting on ${\rho }_{{AB}}$. This means that ${Q}_{A}^{{{\rm{B}}}_{\infty }}({\rho }_{{AB}})={Q}_{A}^{{R}_{F}}({\rho }_{{AB}})$ and we recover the surprisal of measurement recoverability ${Q}_{A}^{{R}_{F}}({\rho }_{{AB}})$ from equation (131). Hence, ${Q}_{A}^{{R}_{F}}({\rho }_{{AB}})$ has an operational interpretation in terms of the minimum distinguishability between ${\rho }_{{AB}}$ and its unilocally broadcast copies in the asymptotic setting. Furthermore, it has been suggested that each ${Q}_{A}^{{{\rm{B}}}_{N}}({\rho }_{{AB}})$ is a valid measure of one-sided QCs in its own right [207], obeying at least requirements (i), (ii) and (iv). These quantities are simple to calculate numerically as the solution of a semidefinite programme, and display a natural hierarchy [206, 207]

$$\begin{eqnarray}&&{Q}_{A}^{{{\rm{B}}}_{2}}({\rho }_{{AB}})\leqslant {Q}_{A}^{{{\rm{B}}}_{N}}({\rho }_{{AB}})\leqslant {Q}_{A}^{{R}_{F}}({\rho }_{{AB}})\leqslant {Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}}),\end{eqnarray} \tag{ 145 }$$

hence providing computable lower bounds to quantum discord.

In summary, the ability to locally broadcast a correlated quantum state is fundamentally (and inversely) linked to the QCs present in the state, in both qualitative and various quantitative senses.

The entanglement activation framework presented in section 3.2.4 provides already an operational setting to interpret measures of QCs in a state ${\rho }_{{AB}}$ in terms of their potential for the creation of entanglement with ancillary systems. Under some conditions, this entanglement can be swapped back to the original system AB [220]. More generally, this means that all the measures ${Q}_{A}^{{E}^{\zeta }}$ and ${Q}_{{AB}}^{{E}^{\zeta }}$ of QCs of the entanglement activation type, some of which are listed in table 6, may be regarded as having an intrinsically operational character [36].

Here we discuss the role of QCs for a different task, namely 'entanglement distribution'. Distributing entanglement between spatially separated laboratories is a key starting point in many quantum information protocols. Consider two laboratories A and B, who want to increase their shared entanglement. From the fundamental paradigm of entanglement, it is clear that such a task cannot be achieved using only LOCC. Instead, it is necessary to send a quantum particle C from one laboratory to the other (we will assume the particle is sent from A to B), i.e. to utilise quantum communication in combination with local operations.

More rigorously, we can look at the state ${\rho }_{{ABC}}$ of the composite system consisting of the two parties A and B and the carrier particle C. It is then relevant to consider two bipartitions of the three parties. Bipartition ${AC}\,:B$ represents the carrier particle at laboratory A, while bipartition $A:{BC}$ represents the carrier particle after having been sent to laboratory B. We then turn to the entanglement ${E}_{{AC}:B}({\rho }_{{ABC}})$ and ${E}_{A:{BC}}({\rho }_{{ABC}})$, which represent the entanglement before and after the carrier particle has been sent (through a noiseless quantum channel), where ${E}_{X:Y}$ is a measure of bipartite entanglement for the bipartition $X\,:Y$. The objective is to achieve ${E}_{A:{BC}}({\rho }_{{ABC}})\gt {E}_{{AC}:B}({\rho }_{{ABC}})$. Laboratory B can then attempt to perform joint operations on both their system and the carrier particle to localise the entanglement onto B and hence increase the entanglement between A and B.

The question is whether we can place a minimum cost on the increase in entanglement ${E}_{A:{BC}}({\rho }_{{ABC}})-{E}_{{AC}:B}({\rho }_{{ABC}})$. One might guess that the cost is related to ${E}_{{AB}:C}({\rho }_{{ABC}})$, i.e. so that the increase in entanglement is limited by the entanglement carried by particle C with the two laboratories. In fact, entanglement across the ${AB}:C$ split turns out not to be necessary, and the task of entanglement distribution can also be accomplished with a separable carrier, as first shown in [221]. Instead, [222, 223] suggest that more general QCs between AB and C, defined with respect to one-sided measurements on C, may represent appropriately the minimum cost. They showed that, when resorting to the geometric measures of QCs (see section 3.2.1), it holds

$$\begin{eqnarray}&&{Q}_{C}^{{G}_{\delta }}({\rho }_{{ABC}})\geqslant {E}_{A:{BC}}^{{G}_{\delta }}({\rho }_{{ABC}})-{E}_{{AC}:B}^{{G}_{\delta }}({\rho }_{{ABC}}),\end{eqnarray} \tag{ 146 }$$

where ${E}_{X:Y}^{{G}_{\delta }}({\rho }_{{ABC}})$ is the corresponding geometric measure of entanglement defined in equation (47) for the bipartition $X\,:Y$. This inequality, which in the case of measures based on the relative entropy represents a strengthening of the standard subadditivity inequality of von Neumann entropy, tells us that QCs on the C side are in general necessary for entanglement distribution. In fact, whenever QCs are not present, the transmission of the carrier particle C effectively reduces to classical communication. Examples of states ${\rho }_{{ABC}}$ were given in [222] for when this bound is tight, and it was argued in [223] that entanglement distribution with separable states may be more efficient than using entangled carriers if local generation of entanglement at the sender laboratory is expensive.

However, the usefulness of a state for entanglement distribution is not determined only by its QCs. In [224] examples of states with nonzero QCs yet that cannot be used for entanglement distribution were provided, and additional conditions to ensure usefulness for this task were derived in terms of the dimension of C and the number of product terms in the decomposition of the separable ${\rho }_{{ABC}}$ along the partition ${AB}\,:C$. This situation was likened to entanglement, whereby only some entangled states are useful for distillation (i.e. the states being not bound entangled) [3]. It was also suggested in [225] that the quantity ${Q}_{C}^{{G}_{\delta }}({\rho }_{{ABC}})$ may not a relevant resource for entanglement distribution because laboratory A initially holds the carrier C and can thus alter the QCs by local operations, potentially making equation (146) a rather loose bound.

Finally, a number of experiments have successfully demonstrated entanglement distribution with separable states in discrete and continuous variable setups [226–228].

Imagine that our two laboratories A and B each sample from discrete random variables, with a joint probability distribution p_ab where $a\in A$ and $b\in B$ are the possible values found at each laboratory, see section 3.5.2 for further details. With N repeated samples, they arrive at a sequence of values $\{({a}_{1},{b}_{1}),({a}_{2},{b}_{2}),\ldots ,({a}_{N},{b}_{N})\}$. However, neither laboratory knows the results of the other. If laboratory B wants to transmit their results to A, how much information must they send? Shannon's noiseless coding theorem tells us that for large N, B can send as little as ${ \mathcal H }(B)$ bits per sample for A to successfully reconstruct the complete sequence [2, 144]. Such a scheme, however, does not exploit any correlations between the two random variables. If A has some prior information on B, i.e. if ${ \mathcal I }(A\,:B)\gt 0$, then can B exploit this to send fewer than ${ \mathcal H }(B)$ bits per sample? The answer is yes, only the partial information that A is missing about B needs to be transmitted, which can be achieved with B sending only ${ \mathcal H }(B| A)={ \mathcal H }({AB})-{ \mathcal H }(A)$ ($\leqslant { \mathcal H }(B)$) bits of information per sample, provided N is large [229]. Importantly, this may be completed without B having any knowledge of the prior information that laboratory A has.

An analogous problem may be studied in the quantum setting. Now consider a quantum source that emits a bipartite pure state ${\left|| ,{\psi }_{i},\rangle \right.\rangle }_{{AB}}$ with probability p_i and distributes this state to laboratories A and B. The laboratories know the statistics of the source and hence the mixed state ${\rho }_{{AB}}={\sum }_{i}{p}_{i}\left|| ,{\psi }_{i},\rangle \right.\rangle {\left.\langle \langle ,{\psi }_{i},| \right|}_{{AB}}$, but not the ensemble $\{{p}_{i},{\left|| ,{\psi }_{i},\rangle \right.\rangle }_{{AB}}\}$ realising it. Suppose that the laboratories sample from this source N times and have the sequence of (unknown) pure states $\{{\left|| ,{\psi }_{{i}_{1}},\rangle \right.\rangle }_{{AB}},{\left|| ,{\psi }_{{i}_{2}},\rangle \right.\rangle }_{{AB}},\ldots ,{\left|| ,{\psi }_{{i}_{N}},\rangle \right.\rangle }_{{AB}}\}$. For each sample of the source, how can B transmit their share of the distributed pure state to A? Both classical and quantum communication (and local operations) are at B's disposal. However, classical communication is a much simpler task than quantum communication, which requires sending of sensitive quantum information down noisy channels. Hence, we allow for unlimited and free classical communication and concern ourselves with finding the most efficient way in terms of quantum communication for B to transmit their share of the unknown state to A. With free classical communication we are able to carry out quantum communication via teleportation [230], using a bank of maximally entangled Bell states ${\left|| ,{\rm{\Phi }},\rangle \right.\rangle }_{{AB}}=\tfrac{1}{\sqrt{2}}({\left|| ,0,\rangle \right.\rangle }_{A}{\left|| ,0,\rangle \right.\rangle }_{B}+{\left|| ,1,\rangle \right.\rangle }_{A}{\left|| ,1,\rangle \right.\rangle }_{B})$ shared between A and B. The question then becomes: what is the minimum entanglement, quantified by shared Bell states (or 'ebits'), required per copy of ${\rho }_{{AB}}$ for B to transmit their state to A?

This question was tackled in [231, 232]. Here it was shown that for any ensemble $\{{p}_{i},{\left|| ,{\psi }_{i},\rangle \right.\rangle }_{{AB}}\}$ realising the mixed state ${\rho }_{{AB}}$ and for $N\to \infty$, the rate of ebit consumption is at least ${\rm{\max }}\{0,{{ \mathcal S }}_{B| A}({\rho }_{{AB}})\}$, with ${{ \mathcal S }}_{B| A}({\rho }_{{AB}})={ \mathcal S }({\rho }_{{AB}})-{ \mathcal S }({\rho }_{A})$ the conditional entropy of equation (88) (not to be confused with the measurement based quantity in equation (89)), representing the partial information needed for B to transmit their state to A. However, it is a curious fact that the quantum conditional entropy can be negative for some ${\rho }_{{AB}}$ (consider, for example, any pure entangled state). In that case, B may transmit their state to A using only LOCC. Even better, it is then possible to create shared ebits at a rate no more than $-{{ \mathcal S }}_{B| A}({\rho }_{{AB}})$. Thus, the negative quantum conditional entropy can be understood as the potential for future quantum communication between A and B in the form of teleportation.

The optimal protocol allowing B to transmit their state to A is called 'quantum state merging', and can be equivalently thought of in the following way [232]. Imagine that a third reference laboratory C is present so that the composite state of the three systems ${\left|| ,\psi ,\rangle \right.\rangle }_{{ABC}}$ is pure. State merging is the task of performing LOCC and optimally drawing from (or contributing to) the bank of shared ebits to transform the state ${\left|| ,\psi ,\rangle \right.\rangle }_{{ABC}}^{\otimes N}$ into the state ${\left|| ,\psi ,\rangle \right.\rangle }_{{ADC}}^{\otimes N}$, where D is an ancilla in laboratory A designed to hold the state of B. The fidelity between ${\left|| ,\psi ,\rangle \right.\rangle }_{{ABC}}^{\otimes N}$ and ${\left|| ,\psi ,\rangle \right.\rangle }_{{ADC}}^{\otimes N}$ must be high, tending to unity as $N\to \infty$, which means that the local state of the reference laboratory C is effectively unchanged throughout the process. Working within this setting is useful in our following exposition on the role of QCs in state merging.

One link with QCs was provided in [233]. They noted the fact that entanglement between A and B is destroyed during the state merging process and so considered the total entanglement consumption ${E}_{A:B}^{f}({\rho }_{{AB}})+{{ \mathcal S }}_{B| A}({\rho }_{{AB}})$, where ${E}_{A:B}^{f}({\rho }_{{AB}})$ is the entanglement of formation between A and B (see table 4). Here, ${E}_{A:B}^{f}({\rho }_{{AB}})$ quantifies the entanglement present before merging (which is entirely consumed), and ${{ \mathcal S }}_{B| A}({\rho }_{{AB}})$ quantifies the entanglement gained or lost after merging. By manipulating equation (97) [176, 177], given that the global state of the tripartite system ABC is pure, one finds that this total entanglement consumption is equal to the mutual information based measure of QCs, i.e. the quantum discord, of ${\rho }_{{BC}}={{\rm{Tr}}}_{A}(\left|| ,\psi ,\rangle \right.\rangle {\left.\langle \langle ,\psi ,| \right|}_{{ABC}})$ with one-sided measurements on C

$$\begin{eqnarray}&&{E}_{A:B}^{f}({\rho }_{{AB}})+{{ \mathcal S }}_{B| A}({\rho }_{{AB}})={Q}_{C}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{BC}}).\end{eqnarray} \tag{ 147 }$$

Hence, the QCs between B and C, with measurements on C, can be quantitatively understood as the total entanglement consumption in quantum state merging. The authors of [233] also considered the corresponding asympotic total entanglement consumption, i.e. 

$$\begin{eqnarray}&&{E}_{A:B}^{f}{\left((,{\rho }_{{AB}},)\right)}_{\infty }+{{ \mathcal S }}_{B| A}({\rho }_{{AB}})={Q}_{C}^{{\tilde{I}}_{{ \mathcal I }}}{\left((,{\rho }_{{BC}},)\right)}_{\infty },\end{eqnarray} \tag{ 148 }$$

where ${E}_{A:B}^{f}{({\rho }_{{AB}})}_{\infty }$ is the entanglement cost [3], thus providing an interpretation to the regularised measure of QCs given in equation (101).

These findings give an operational motivation for the asymmetry present in the one-sided measures of QCs, that is, in general ${Q}_{C}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{BC}})\ne {Q}_{B}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{BC}})$ because the total entanglement consumed for B to merge with A is not necessarily the same as for C to merge with A. A link to dense coding was also provided in [233]. In dense coding, one can use quantum communication to transmit classical information at a faster rate than with classical communication, and it was shown that ${Q}_{C}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AC}})-{Q}_{C}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{BC}})$, with ${\rho }_{{AC}}={{\rm{Tr}}}_{B}(\left|| ,\psi ,\rangle \right.\rangle {\left.\langle \langle ,\psi ,| \right|}_{{ABC}})$, quantifies exactly the difference in the quantum advantage of dense coding for C to communicate with A as opposed to B.

An alternative operational view of QCs in terms of state merging was found in [234]. Here it was shown that the regularised QCs ${Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}{({\rho }_{{AB}})}_{\infty }$ of equation (101) represent the minimum increase in the cost of state merging (in terms of the rate of ebit consumption) if one first performs local measurements on each copy of A. Specifically, if ${{ \mathcal S }}_{B| A}({\rho }_{{AB}})$ is the cost of state merging before and ${{ \mathcal S }}_{B| A}{({\tilde{{\rm{\Pi }}}}_{A}[{\rho }_{{AB}}])}_{\infty }$ is the cost of state merging after each LGM on A, then

$$\begin{eqnarray}&&\mathop{\inf }\limits_{{\tilde{{\rm{\Pi }}}}_{A}}{{ \mathcal S }}_{B| A}{\left((,{\tilde{{\rm{\Pi }}}}_{A}[{\rho }_{{AB}}],)\right)}_{\infty }-{{ \mathcal S }}_{B| A}({\rho }_{{AB}})={Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}{\left((,{\rho }_{{AB}},)\right)}_{\infty }.\end{eqnarray} \tag{ 149 }$$

This operational link was then extended in [235, 236] by considering what is known as the fully quantum Slepian–Wolf protocol [237], which is the 'mother of all protocols' since it is so general that it contains as special cases quantum state merging, dense coding, teleportation and entanglement distillation [2, 205]. Again, the regularised QCs ${Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}{({\rho }_{{AB}})}_{\infty }$ represent the minimum drop in performance of the fully quantum Slepian–Wolf protocol after local measurements on A.

Finally, in [238] the primitive of 'information concentration' was studied, a variation of state merging in which two parties A and B aim to maximise their mutual information by means of LOCC operations performed by B and a third party C. The figure of merit for this protocol was found to be a tripartite generalisation of the quantum discord.

The quantum state redistribution protocol consists of a sender holding two systems A and C, a receiver holding instead only one system B, and a reference system R that is inaccessible to both sender and receiver, all sharing a four-partite pure state $| {\psi }_{{ACBR}}\rangle$ [239]. The objective is to redistribute the quantum information in such a way that is instead the receiver who holds C, while retaining the purity of the overall four-partite state. In order to accomplish this task, sender and receiver are only allowed to perform local quantum operations, consume or generate ebits, and the sender can give only qubits to the receiver. More precisely, contrarily to the quantum teleportation protocol [230], they are not allowed to classically communicate beyond what can be encoded in qubits. It turns out that this transfer can occur perfectly in the asymptotic limit of many copies provided that the amount of qubits that the sender gives to the receiver, also called communication cost, is at least half the conditional quantum mutual information ${{ \mathcal I }}_{B}({\rho }_{{CBR}})$ of the reduced state ${\rho }_{{CBR}}={{\rm{Tr}}}_{A}(| {\psi }_{{ACBR}}\rangle \langle {\psi }_{{ACBR}}| )$ between C and R with respect to B; see [239] for more details about the optimal protocol. Interestingly, this protocol is self-dual under time reversal, i.e., it can be reversed by generating the same amount of entanglement and spending the same communication cost. In particular, switching A and B has no effect on the communication cost, indeed for any pure state $| {\psi }_{{ACBR}}\rangle$ it happens that ${{ \mathcal I }}_{B}({\rho }_{{CBR}})={{ \mathcal I }}_{A}({\rho }_{{ACR}})$, with ${\rho }_{{ACR}}={{\rm{Tr}}}_{B}(| {\psi }_{{ACBR}}\rangle \langle {\psi }_{{ACBR}}| )$. We can thus think of the quantity ${{ \mathcal I }}_{B}({\rho }_{{CBR}})={{ \mathcal I }}_{A}({\rho }_{{ACR}})$ as twice the minimal communication cost to transfer C between A and B, regardless of the direction of the transfer, while retaining the purity of the global four-partite state $| {\psi }_{{ACBR}}\rangle$.

The quantum state redistribution protocol provides us with another operational interpretation for the quantum discord, as first highlighted in [240]. Let us consider two laboratories holding, respectively, the quantum systems A and B, which share the bipartite quantum state ${\rho }_{{AB}}$. Suppose that a maximally informative LGM ${\tilde{{\rm{\Pi }}}}_{A}$ is performed on A, with $A^{\prime}$ denoting the corresponding classical output. As already mentioned in section 3.9.1, such LGM can be written as a unitary ${U}_{A\to A^{\prime} C}^{{\tilde{{\rm{\Pi }}}}_{A}}$ from A to the composite system $A^{\prime} C$ followed by discarding C, i.e.,

$$\begin{eqnarray}&&{\tilde{{\rm{\Pi }}}}_{A}[{\rho }_{{AB}}]={{\rm{Tr}}}_{C}\left((,{U}_{A\to A^{\prime} C}^{{\tilde{{\rm{\Pi }}}}_{A}}[{\rho }_{{AB}}],)\right).\end{eqnarray} \tag{ 150 }$$

Moreover, we also know that the quantum discord ${Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}})$ between A and B, equation (132), is exactly given by the conditional quantum mutual information ${{ \mathcal I }}_{{A}^{\prime }}({\rho }_{A^{\prime} {CB}})$ of the state ${\rho }_{A^{\prime} {CB}}={U}_{A\to A^{\prime} C}^{{\tilde{{\rm{\Pi }}}}_{A}}[{\rho }_{{AB}}]$ between C and B with respect to $A^{\prime}$, i.e.,

$$\begin{eqnarray}&&{Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}})={{ \mathcal I }}_{{A}^{\prime }}\left((,{\rho }_{A^{\prime} {CB}},)\right).\end{eqnarray} \tag{ 151 }$$

Now, let R be a reference system purifying the state ${\rho }_{A^{\prime} {CB}}$, i.e., such that ${\rho }_{A^{\prime} {CB}}={{\rm{Tr}}}_{R}(| {\psi }_{A^{\prime} {CBR}}\rangle \langle {\psi }_{A^{\prime} {CBR}}| )$. By looking at equation (150), we can then think of the measurement ${\tilde{{\rm{\Pi }}}}_{A}$ as a process whereby the environment C is lost and given to the reference R. One can then ask the following question. What is the optimal quantum communication cost needed to send system C from R to laboratory A in such a way that the action of the LGM ${\tilde{{\rm{\Pi }}}}_{A}$ can be undone? As already mentioned, this is given by half the conditional quantum mutual information ${{ \mathcal I }}_{{A}^{\prime }}\left((,{\rho }_{A^{\prime} {CB}},)\right)$ between C and B with respect to $A^{\prime}$, by assuming that laboratory B plays no role in the protocol. Therefore, equation (151) provides us with the promised operational interpretation of quantum discord as twice the optimal communication cost needed to restore the environment of a measurement so that it is no longer lost, i.e., to restore the coherence lost in a measurement [240]. Due to the symmetry under time reversal of such optimal protocol, the quantum discord between A and B can also be seen as the communication cost needed to send system C back to the reference R, i.e., as the amount of quantum information lost in the measurement process [240], which neatly captures the physical motivation behind the original definition of discord [6].

Building upon this operational interpretation, in [241] it has been shown that the quantum discord ${Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}})$ of a bipartite quantum state ${\rho }_{{AB}}$ is equal to the minimal rate of noise that one needs to apply to subsystem A in such a way that: (i) the resulting state ${\tilde{\rho }}_{{AB}}$ is locally recoverable after an LGM acts on A and (ii) the post-measurement state corresponding to ${\tilde{\rho }}_{{AB}}$ after such an LGM is indistinguishable from the post-measurement state corresponding to ${\rho }_{{AB}}$ after a maximally informative LGM. A state satisfying properties (i) and (ii) is also said to be approximately einselected, where 'einselection' is the abbreviation for environment-induced superselection and is a process whereby the interaction between a quantum system and the environment is such that only the eigenstates corresponding to particular observables of the system, so-called pointer states, persist in the system [37]. In other words, quantum discord is equal to the optimal cost of simulating einselection. This perhaps stands as a physical interpretation of quantum discord that is even more in line with its own original definition as a measure of the decrease of correlations after einselection is complete [6].

In quantum teleportation [230, 242], the objective is to transmit an unknown quantum state from laboratory A to laboratory B. To achieve such a feat, one must make use of classical communication and some distributed entanglement. In fact, to teleport a qubit it is necessary and sufficient to use two bits of classical communication and one ebit [230]. Instead, in 'remote state preparation', the objective is to transmit a known qubit state. It was shown by [243, 244] that in the many copy setting, this is achievable with the use of one classical bit of communication and one shared ebit per copy, and that in general it is possible to trade-off these two quantities: one can pay with more classical bits to save on ebits, and vice versa.

One special case is remote state preparation of the qubit states of the form $\left|| ,\psi ,\rangle \right.\rangle =\tfrac{1}{\sqrt{2}}(\left|| ,0,\rangle \right.\rangle +{{\rm{e}}}^{{\rm{i}}\phi }\left|| ,1,\rangle \right.\rangle )$, which lie on the equator of the Bloch sphere, as discussed in [245, 246]. In this case, remote state preparation can be performed without resorting to the many copy limit. Consider the shared ebit $\tfrac{1}{\sqrt{2}}(\left|| ,01,\rangle \right.\rangle -\left|| ,10,\rangle \right.\rangle )$, which can be equivalently written (up to a global phase) as $\tfrac{1}{\sqrt{2}}(\left|| ,\psi {\psi }^{\perp },\rangle \right.\rangle -\left|| ,{\psi }^{\perp }\psi ,\rangle \right.\rangle )$, where $\left|| ,{\psi }^{\perp },\rangle \right.\rangle =\tfrac{1}{\sqrt{2}}(\left|| ,0,\rangle \right.\rangle -{{\rm{e}}}^{{\rm{i}}\phi }\left|| ,1,\rangle \right.\rangle )$ is the qubit state orthogonal to $\left|| ,\psi ,\rangle \right.\rangle$. If A performs an LPM on their half of the ebit in the basis $\{\left|| ,\psi ,\rangle \right.\rangle ,{\left|| ,\psi ,\rangle \right.\rangle }^{\perp }\}$, then it is clear that B will either get the state $\left|| ,{\psi }^{\perp },\rangle \right.\rangle$, if the LPM by A resulted in $\left|| ,\psi ,\rangle \right.\rangle$, or the state $\left|| ,\psi ,\rangle \right.\rangle$, if A found $\left|| ,{\psi }^{\perp },\rangle \right.\rangle$. If B has $\left|| ,{\psi }^{\perp },\rangle \right.\rangle$, then $\left|| ,\psi ,\rangle \right.\rangle$ can be retrieved by simply applying a π rotation around the z axis of the Bloch sphere, i.e. $\left|| ,\psi ,\rangle \right.\rangle ={\sigma }_{3}\left|| ,{\psi }^{\perp },\rangle \right.\rangle$, where ${\sigma }_{3}$ is the third Pauli matrix. However, B can only know whether to apply the rotation from the measurement outcome of A, and hence remote state preparation of equatorial states in this single copy case requires one shared ebit and one bit of classical communication.

If one has access to an ebit as a resource in the above protocol, then remote state preparation can be achieved perfectly. Instead, if one uses a more general resource state, such as a mixed state of two qubits ${\rho }_{{AB}}$, then an error is introduced into the remote state preparation. Consider the state to be prepared with Bloch vector $\vec{m}$ lying on the equatorial plane, and the resultant state after the aforementioned protocol with Bloch vector $\vec{n}$. In [103], a quadratic payoff function ${P}_{\vec{m},\vec{n}}^{2}:= {(\vec{m}\cdot \vec{n})}^{2}$, which captures how well the target $\vec{m}$ and the actual $\vec{n}$ overlap and is linked to the fidelity between the corresponding states, was adopted to quantify the performance of the protocol. Precisely, for a given resource state ${\rho }_{{AB}}$, a figure of merit for remote state preparation can be defined by first maximising ${P}_{\vec{m},\vec{n}}$ over all output states $\vec{n}$ (or equivalently all possible LPMs performed by A), then averaging the result over all pure states $\vec{m}$ on the equator, and finally minimising the latter quantity over all possible choices of the reference north pole state on the Bloch sphere, since the orientation of the axes can be arbitrary, as long as this is pre-agreed between A and B. The resulting quantity, that we denote by ${\overline{P}}_{A}^{2}({\rho }_{{AB}})$, can be interpreted as the remote state preparation fidelity achievable with the shared state ${\rho }_{{AB}}$ in a worst case scenario. Interestingly, it was shown in [25] that, for a class of two-qubit states ${\rho }_{{AB}}$ (i.e. the Bell diagonal states), this quantity coincides with (twice) the Hilbert–Schmidt based geometric measure of QCs,

$$\begin{eqnarray}&&{\overline{P}}_{A}^{2}({\rho }_{{AB}})=2{Q}_{A}^{{G}_{2}}({\rho }_{{AB}}),\end{eqnarray} \tag{ 152 }$$

thus apparently providing an operational interpretation to the latter, even though in a specialised setting. The theoretical result was supported experimentally using a photonic implementation, where certain separable states with nonzero QCs were shown to perform remote state preparation better than some entangled states with smaller QCs, according to the figure of merit given by equation (152).

A refinement of this protocol has been presented in [247], where it is argued, however, that the figure of merit ${\overline{P}}_{A}^{2}({\rho }_{{AB}})$ can be misleading, since it can be surpassed simply by employing the trivial protocol of B randomly preparing a pure equatorial state, regardless of the communication from A and the shared quantum resource ${\rho }_{{AB}}$. Instead, the suggested figure of merit should derive from the (non-quadratic) payoff-function ${P}_{\vec{m},\vec{n}}=\vec{m}\cdot \vec{n}$. Generalisations of the encoding and decoding strategies of A and B are also outlined in [247]. Instead of an LPM on the shared resource ${\rho }_{{AB}}$, A can perform a more general two-outcome LGM, and send the result using one bit of classical communication to B. B is then allowed to utilise any local operations to recover the best approximation of $\vec{m}$. In this more general setting, it is proved that it is impossible to use a shared separable state to outperform an entangled state in remote state preparation, hence providing a very critical analysis of the role of QCs beyond entanglement in this communication task. However, their role can be resurrected when certain reasonable restrictions are placed upon the local operations that B is able to perform when trying to recover $\vec{m}$. Two important cases are highlighted where it is possible for quantumly correlated separable states to outperform entangled ones: (i) B does not share a local Bloch reference frame with A, and (ii) B is restricted to unital operations. Even stronger, under condition (ii) and in the case of the shared resource state ${\rho }_{{AB}}$ being Bell diagonal, the remote state preparation figure of merit recovers again a link to the Hilbert–Schmidt based geometric measure of QCs.

A further criticism of the role of QCs in remote state preparation was given by [248]. Here, they discriminate between the QCs that can be created by local operations and those that cannot, and it is argued that only the latter should be considered. In this sense, it is then shown that there are quantumly correlated states that are useless for remote state preparation, while there are states without QCs that can be made useful. Furthermore, it is discussed in [249] that an increase in QCs does not necessarily imply an increase in the quality of remote state preparation, even though in some cases a quantitative link is found between the fidelity of remote state preparation using noisy cluster states and the negativity of quantumness ${Q}_{A}^{{E}^{N}}$ defined in section 3.6.2.

Finally, in [102] a version of remote state preparation is discussed for a system consisting of a qubit and a harmonic oscillator. It is found that the geometric measure ${Q}_{A}^{{G}_{2}}({\rho }_{{AB}})$, where A is the qubit, can provide a lower bound to the payoff of remote state preparation under certain restrictions. These restrictions are on the receiving oscillator B: if B is restricted to only local unitary corrections following the measurement by A then ${Q}_{A}^{{G}_{2}}({\rho }_{{AB}})$ presents a limit on the performance of remote state preparation, whereas if B is free to perform arbitrary (non-unitary) local operations then ${Q}_{A}^{{G}_{2}}({\rho }_{{AB}})$ becomes no longer relevant, and the payoff reduces to an entanglement measure. This lends support to the findings of [247].

The idea behind 'quantum key distribution' is to harness quantum mechanics to tackle the classically problematic task of distributing cryptographic keys securely, without a third party eavesdropper discovering what has been communicated [250]. In fact, the distributors and receivers of a key encoded in a quantum system can detect an attempt at eavesdropping (up to a sensitivity threshold) due to the disturbance induced by measurements on their system. From the outset, this problem seems tailored to utilising the resource of QCs and in particular the inherent non-orthogonality of bipartite states with nonzero QCs. Consequently, it was shown in [251] that quantumly correlated states are necessary for quantum key distribution in the device-dependent setting—which corresponds to the distributing and receiving parties being able to trust their devices.

In the practically relevant case of distribution of secure keys over a lossy channel with transmissivity η, such as a free-space link or an optical fibre, the secret key capacity K has been very recently determined and found to coincide with the maximum quantum discord that can be distributed to the remote parties through such a lossy channel [252]. In the following we briefly explain this result, which connects QCs to the ultimate limits of quantum communication in continuous variable systems. Further details are available in [252].

In general, a quantum cryptographic protocol [250] involves two distant laboratories, a sender A and a receiver B, separated by a quantum channel. The sender prepares ensembles of (non-orthogonal) input states and transmits them to the receiver, who can measure the outputs. They can resort to unlimited classical communication, which allows them to extract a key through error correction and privacy amplification. The secret key capacity K is defined as the maximum number of secret key bits per channel use which can be distributed over the quantum channel, obtained upon maximising over all possible input states of the sender A and all possible output measurements by the receiver B (in general, the input-output local operations may be adaptive, i.e., assisted by two-way classical communication [252]). An achievable protocol can be represented in terms of N ebits of entanglement distributed over the channel, followed by LOCC on both parties. Let us denote by ${\rho }_{{AB}}^{\infty }$ the output state of A and B, in the limit $N\to \infty$ (i.e., in the limit of an ideal maximally entangled distributed state, also known as an EPR state [17]). The result in [252] then shows that, in the important practical case of a bosonic lossy channel with transmissivity η, it holds

$$\begin{eqnarray}&&K(\eta )={Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}}^{\infty })=-\,{\mathrm{log}}_{2}(1-\eta ),\end{eqnarray} \tag{ 153 }$$

where in particular Gaussian LGMs are found to be optimal to calculate the quantum discord ${Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}}^{\infty })$ of the asymptotic two-mode Gaussian state ${\rho }_{{AB}}^{\infty }$, due to the results of [120, 174].

The further exploration of the role and power of QCs in quantum key distribution protocols certainly has potential [251–253], but its scope exceeds the present review, and we point the interested reader to [250, 254] for an overview on quantum cryptography.

Let us begin in the classical setting by considering a generic measure of correlations between two random variables A and B. We might expect that this measure cannot increase by more than m bits if one were to carry out m bits of classical communication. Indeed, this intuitive property holds in particular for the mutual information ${ \mathcal I }(A\,:B)$ of equation (83) [148]. How does this feature extends to the quantum setting, where quantum communication is also possible? Suppose we apply a scheme of local operations and either n qubits of quantum communication or $2n$ bits of classical communication, acting on a bipartite state ${\rho }_{{AB}}$ shared between laboratory A and B. We may now require that a quantifier of correlations cannot increase by more than $2n$ bits after this scheme, since we can view each qubit of quantum communication as effectively two bits of classical communication due to dense coding. This property holds indeed for the quantum mutual information ${ \mathcal I }({\rho }_{{AB}})$ of equation (87) [147].

Interestingly, this property does not hold for the classical correlations ${\tilde{{ \mathcal J }}}_{{AB}}$ of equation (94), representing the correlations available to laboratories A and B through local measurements. Consider the bipartite $(2N+1)$-qubit state ${\rho }_{{AB}}$ with A having N qubits and B having $N+1$,

$$\begin{eqnarray}&&{\rho }_{{AB}}=\displaystyle \frac{1}{{2}^{N+1}}\displaystyle \sum _{i=0}^{{2}^{N}-1}\displaystyle \sum _{j=0}^{1}{\left((,{U}_{j}\left|| ,i,\rangle \right.\rangle \left.\langle \langle ,i,| \right|{U}_{j}^{\dagger },)\right)}_{A}\otimes {\left((,\left|| ,i,\rangle \right.\rangle \left.\langle \langle ,i,| \right|\otimes \left|| ,j,\rangle \right.\rangle \left.\langle \langle ,j,| \right|,)\right)}_{B},\end{eqnarray} \tag{ 154 }$$

where ${U}_{0}={\mathbb{I}}$ and $| \left\langle \langle ,i^{\prime} | {U}_{1}| i,\rangle \right\rangle {| }^{2}=1/{2}^{N}$, i.e. with $\{{U}_{1}\left|| ,i,\rangle \right.\rangle \}$ producing a mutually unbiased basis with respect to the basis $\{\left|| ,i,\}\right\}$ [255]. For example, we may fix $\{\left|| ,i,\rangle \right.\rangle \}$ to be the computational basis and choose ${U}_{1}={H}^{\otimes N}$, with H the Hadamard gate. This state may be prepared by having laboratory B pick a random N-bit string with label i and sending to A either ${U}_{0}\left|| ,i,\rangle \right.\rangle =\left|| ,i,\rangle \right.\rangle$ or ${U}_{1}\left|| ,i,\rangle \right.\rangle$, again at random based on the value $j\in \{0,1\}$. The total correlations are here ${ \mathcal I }({\rho }_{{AB}})=N$, but the classical correlations are ${\tilde{{ \mathcal J }}}_{{AB}}({\rho }_{{AB}})=N/2$. Hence, some of the correlations are locked, i.e. inaccessible to A and B through local measurements. However, if B sends the result j to laboratory A, then A can reverse U_j. Laboratory A then measures in the computational basis so that they together share the N-bit string i, and therefore $N+1$ bits of classical correlations (including the communicated bit). They have thus unlocked $N/2$ bits of classical correlations with only one bit of classical communication.

This curious phenomenon is a truly quantum effect, yet can occur in separable states like in equation (154). Clearly then, entanglement is not a figure of merit in locking of classical correlations, so what about more general QCs? The first hints that QCs play a role were given in [143, 256]. There it was noted that both the two-sided mutual information based measure of QCs ${Q}_{{AB}}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}})$ and the measurement induced disturbance (see section 3.5.2) are equal to $N/2$ in the above example, exactly the amount of locked classical correlations.

The link with QCs was placed on a firmer footing in [257]. In their setting, laboratory B samples from a random source of numbers i and wants to send the result to A, but has to use a noisy quantum channel so that A eventually receives ${\rho }_{A}^{(i)}$. Together A and B share a quantum–classical state ${\rho }_{{AB}}\in {C}_{B}$, just like the one in equation (154), and the objective is for A to infer the value of i. The classical correlations tell us how well A can do this. If laboratory B can then send a key to reveal the value of i to A, it turns out that the amount of classical correlations unlocked in doing so is exactly the quantum discord ${Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}({\rho }_{{AB}})$ [257]. Furthermore, in the asymptotic limit of many copies of ${\rho }_{{AB}}$, it was shown that the regularised QCs measure ${Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}{({\rho }_{{AB}})}_{\infty }$ of equation (101) quantifies the quantum advantage for A to successfully infer the message from B when compared to a corresponding classical protocol, in terms of the key length per copy required to unlock the correlations; ${Q}_{A}^{{\tilde{I}}_{{ \mathcal I }}}{({\rho }_{{AB}})}_{\infty }$ then also gives the amount of classical correlations unlocked per copy in the quantum protocol [257].

Quantum metrology studies how to harness quantum mechanical effects to enhance the precision in estimating physical quantities not amenable to a direct observation [258, 262, 263]. A relevant class of problems in quantum metrology can be formalised in terms of phase estimation in an interferometric setup, which is akin to the scheme in figure 3(c). The estimation procedure then consists of the following steps. An input state ${\rho }_{{AB}}$ enters a two-arm channel, in which the idler subsystem B is unaffected while the probe subsystem A undergoes a local unitary transformation U_A, so that the output density matrix can be written as ${\rho }_{{AB}}^{\varphi }:= {({U}_{A}\otimes {{\mathbb{I}}}_{B}){\rho }_{{AB}}({U}_{A}\otimes {{\mathbb{I}}}_{B})}^{\dagger }$, with ${U}_{A}={{\rm{e}}}^{-{\rm{i}}\varphi {H}_{A}}$, where φ is the parameter one aims to estimate and H_A is a (non-degenerate) Hamiltonian generator of the local unitary dynamics. Finally, the information on φ is recovered by constructing an unbiased estimator $\hat{\varphi }$ obtained by classical processing of the data resulting from (possibly joint) measurements of suitable observables on the output state ${\rho }_{{AB}}^{\varphi }$.

For any input state ${\rho }_{{AB}}$ and known generator H_A, and provided n i.i.d. iterations of the probing procedure are implemented, the maximum achievable precision is determined theoretically by the quantum Cramér-Rao bound [264], which says that the mean square error ${{\rm{\Delta }}}^{2}\hat{\varphi }$ of any unbiased estimator scales as