On state versus channel quantum extension problems: exact results for $U\;\otimes \;U\;\otimes \;U$ symmetry

The most familiar identification between linear maps on operator space and bipartite operators is the Choi isomorphism [21, 25]. Despite having useful properties, the Choi isomorphism will not be suitable for the identification we seek. Instead, we shall develop an operationally motivated variant of the related Jamiołkowski isomorphism [20], that will serve as our bridge between linear maps and bipartite operators.

Let $\mathcal{L}({{\mathcal{H}}_{A}},{{\mathcal{H}}_{B}})$ be the set of linear maps, or 'superoperators', from $\mathcal{B}({{\mathcal{H}}_{A}})$ to $\mathcal{B}({{\mathcal{H}}_{B}})$. The Choi isomorphism, here denoted $\mathcal{C}$, identifies each map $\mathcal{M}\in \mathcal{L}({{\mathcal{H}}_{A}},{{\mathcal{H}}_{B}})$ with the state resulting from the mapʼs action on one subsystem of a maximally entangled state:

$$\begin{eqnarray}&&\mathcal{C}(\mathcal{M})\equiv {{M}_{C}}=[{{\mathcal{I}}_{A}}\otimes \mathcal{M}](\mid {{\Phi }^{+}}\rangle \langle {{\Phi }^{+}}\mid )=\frac{1}{{{d}_{A}}}\mathop{\sum }\limits_{ij}|i\rangle \langle \;j|\otimes \mathcal{M}(|i\rangle \langle \;j|),\end{eqnarray} \tag{ 1 }$$

where ${{\mathcal{I}}_{A}}$ is the identity map on $\mathcal{B}({{\mathcal{H}}_{A}})$, $|{{\Phi }^{+}}\rangle ={{\sum }_{i}}|ii\rangle /\sqrt{{{d}_{A}}}$ and ${{d}_{A}}={\rm dim}({{\mathcal{H}}_{A}})$. We note that ${{d}_{A}}|{{\Phi }^{+}}\rangle \langle {{\Phi }^{+}}|={{V}^{{{T}_{A}}}}$, where T_A denotes partial transposition on subsystem A and V is the swap operator on ${{\mathcal{H}}_{A}}\otimes {{\mathcal{H}}_{A}}$, defined by $V={{\sum }_{i,j}}|ij\rangle \langle ji|$ with respect to any orthonormal basis $\{|i\rangle \}$ on ${{\mathcal{H}}_{A}}$. The transformation is an isomorphism in that it preserves the positivity of the objects it maps to and from; namely, quantum channels (CPTP maps) are mapped to quantum states (positive trace-one operators). Consequently, $\mathcal{C}$ is a useful diagnostic tool for determining whether a map is CP.

The Choi map, however, inherits the local basis dependence of the state $|{{\Phi }^{+}}\rangle$. In fact, the structure of the isomorphism in equation (1) naturally identifies a continuous family of maps generated by $(\mathbb{I}\otimes U)|{{\Phi }^{+}}\rangle$ and parameterized by the set of unitary transformations. For this reason, one should rather understand $\mathcal{C}$ to connect each CP linear map to a whole family of positive bipartite operators, and analogously for non-CP linear maps. Thus, while the Choi isomorphism is an appropriate tool for characterizing the preservation of the positivity constraint, we seek a bonafide identification which bears an operational significance—linking, more strongly, the properties of the linear map with those of the resulting bipartite operator.

The Jamiołkowski isomorphism, often conflated with the Choi isomorphism, is another means of identifying linear maps on operator space with bipartite operators. It differs from the Choi isomorphism in that the swap operator on ${{\mathcal{H}}_{A}}\otimes {{\mathcal{H}}_{A}}$ is used in place of $|{{\Phi }^{+}}\rangle \langle {{\Phi }^{+}}|$ to generate the identification. That is:

$$\begin{eqnarray}&&\mathcal{J}(\mathcal{M})\equiv {{M}_{J}}=[{{\mathcal{I}}_{A}}\otimes \mathcal{M}](V)=\mathop{\sum }\limits_{ij}|i\rangle \langle j|\otimes \mathcal{M}(|j\rangle \langle i|).\end{eqnarray} \tag{ 2 }$$

Notably, the difference between $\mathcal{C}$ and $\mathcal{J}$ has been stressed in [15]; wherein, the authors use the Jamiołkowski isomorphism to translate any quantum channel ${{\mathcal{M}}_{B|A}}\;:\mathcal{B}({{\mathcal{H}}_{A}})\to \mathcal{B}({{\mathcal{H}}_{B}})$ into a causal conditional state, ${{\varrho }_{B|A}}\equiv \mathcal{J}({{\mathcal{M}}_{B|A}})\in \mathcal{B}({{\mathcal{H}}_{A}}\otimes {{\mathcal{H}}_{B}})$.

The desideratum for the identification we seek calls for an operational definition of states and channels exhibiting the same statistical correlations. In the case of a bipartite state, we consider such correlations to be given by the set of probabilities $p(i,j)$ for the joint outcomes of all local POVM pairs $\{E_{i}^{A}\}$ and $\{F_{j}^{B}\}$, on A and B, respectively. The analogous notion for a channel is slightly more involved. Clearly, the POVMs on system A and system B must correspond to measurements before and after the application of the channel, respectively. But, to generate a joint probability distribution with these, we must introduce some initial, pre-measurement state ${{\rho }_{A}}$ of A informing the measurement statistics of $\{E_{i}^{A}\}$. The only unbiased choice is ${{\rho }_{A}}=\mathbb{I}/{{d}_{A}}$, corresponding to no prior information about the input system. Putting this together, we have the following:

Definition 2.2. A bipartite quantum state $\rho \in \mathcal{B}({{\mathcal{H}}_{A}}\otimes {{\mathcal{H}}_{B}})$ and a quantum channel $\mathcal{M}\;:\mathcal{B}({{\mathcal{H}}_{A}})\to \mathcal{B}({{\mathcal{H}}_{B}})$ exhibit the same correlations if for every pair of local POVMs, $\{E_{i}^{A}\}$ and $\{F_{j}^{B}\}$, the likelihood of each joint outcome $p(i,j)$ on ρ is equal to the likelihood of obtaining outcome i given state $\mathbb{I}/{{d}_{A}}$ on A and outcome j after $\mathcal{M}$ has acted.

We now show that a rescaled version of $\mathcal{J}$ provides the desired identification:

Proposition 2.3. A bipartite state $\rho \in \mathcal{B}({{\mathcal{H}}_{A}}\otimes {{\mathcal{H}}_{B}})$ and a quantum channel $\mathcal{M}\;:\mathcal{B}({{\mathcal{H}}_{A}})\to \mathcal{B}({{\mathcal{H}}_{B}})$ exhibit the same correlations if and only if $\mathcal{J}(\mathcal{M})/{{d}_{A}}=\rho$.

Proof. First we calculate the statistical correlations for ρ and then $\mathcal{M}$. For the state ρ, the statistical correlations for the local POVMs $\{E_{i}^{A}\}$ and $\{F_{j}^{B}\}$ are simply

$$\begin{eqnarray}&&p(i,j)={\rm tr}(E_{i}^{A}\otimes F_{j}^{B}\rho ).\end{eqnarray} \tag{ 3 }$$

In the channel case, we calculate the statistical correlations using the conditional probability, $p(i,j)=p(i)p(j|i)$. The first factor is $p(i)={\rm tr}(E_{i}^{A}\mathbb{I}/{{d}_{A}})={\rm tr}(E_{i}^{A})/{{d}_{A}}$. To calculate the second factor, we need an expression for the state of the system conditioned on outcome i. Most generally, the post-measurement state may be written as

$$\begin{eqnarray*}&&{{\rho }_{i}}=\frac{\sqrt{E_{i}^{A}}U\;\mathbb{I}/{{d}_{A}}{{U}^{\dagger }}\sqrt{E_{i}^{A}}}{{\rm tr}(\sqrt{E_{i}^{A}}U\;\mathbb{I}/{{d}_{A}}{{U}^{\dagger }}\sqrt{E_{i}^{A}})}=\frac{E_{i}^{A}}{{\rm tr}(E_{i}^{A})},\end{eqnarray*}$$

from which it follows that

$$\begin{eqnarray*}&&p(j|i)={\rm tr}[F_{j}^{B}\mathcal{M}({{\rho }_{i}})]={\rm tr}[F_{j}^{B}\mathcal{M}(E_{i}^{A})]/{\rm tr}(E_{i}^{A}).\end{eqnarray*}$$

Putting these together, the statistical correlations for a channel are given by

$$\begin{eqnarray*}&&p(i,j)=\frac{1}{{{d}_{A}}}{\rm tr}[F_{j}^{B}\mathcal{M}(E_{i}^{A})].\end{eqnarray*}$$

This above expression may be rewritten using the Jamiołkowski map as follows:

$$\begin{eqnarray}\begin{array}{rcl} \frac{1}{{{d}_{A}}}{\rm tr}[F_{j}^{B}\mathcal{M}(E_{i}^{A})] & = & \frac{1}{{{d}_{A}}}\mathop{\sum }\limits_{i,j}\langle j|E_{i}^{A}|i\rangle {\rm tr}[F_{j}^{B}\mathcal{M}\left( |j\rangle \langle i| \right)] \\ {} & = & \frac{1}{{{d}_{A}}}\mathop{\sum }\limits_{i,j}{\rm tr}[F_{j}^{B}(|i\rangle \langle j|E_{i}^{A})\otimes \left( \mathcal{M}\left( |j\rangle \langle i| \right) \right)] \\ {} & = & \frac{1}{{{d}_{A}}}\mathop{\sum }\limits_{i,j}{\rm tr}[F_{j}^{B}|i\rangle \langle j|\otimes \mathcal{M}\left( |j\rangle \langle i| \right)E_{i}^{A}] \\ {} & = & {\rm tr}[E_{i}^{A}\otimes F_{j}^{B}\mathcal{J}(\mathcal{M})/{{d}_{A}}]. \\ \end{array}\end{eqnarray} \tag{ 4 }$$

Comparing equations (3) and (4), the 'if' of the proposition follows immediately. For the 'only if' direction, we use the fact that the real span of all joint POVM elements is the Hermitian sector of the space $\mathcal{B}({{\mathcal{H}}_{A}}\otimes {{\mathcal{H}}_{B}})$, and hence

$$\begin{eqnarray*}&&{\rm tr}(E_{i}^{A}\otimes F_{j}^{B}\rho )={\rm tr}[E_{i}^{A}\otimes F_{j}^{B}\mathcal{J}(\mathcal{M})/{{d}_{A}}]\end{eqnarray*}$$

can hold for all POVMs and their elements only if $\rho =\mathcal{J}(\mathcal{M})/{{d}_{A}}$.

Because of its operational significance and use in this paper, we provide a name for the above correspondence. Formally, we define the homocorrelation map, $\mathcal{H}$, by letting

$$\begin{eqnarray}&&\mathcal{H}(\mathcal{M})\equiv {{M}_{H}}=[{{\mathcal{I}}_{A}}\otimes \mathcal{M}](V/{{d}_{A}})=\frac{1}{{{d}_{A}}}\mathcal{J}(\mathcal{M}).\end{eqnarray} \tag{ 5 }$$

From equation (4), we may take the defining property of the homocorrelation map to be

$$\begin{eqnarray}&&{\rm tr}[\mathcal{H}(\mathcal{M})A\otimes B]=\frac{1}{{{d}_{A}}}{\rm tr}[\mathcal{M}(A)B],\quad \forall A\in \mathcal{B}({{\mathcal{H}}_{A}}),B\in \mathcal{B}({{\mathcal{H}}_{B}}).\end{eqnarray} \tag{ 6 }$$

The relationships among these various transformations are depicted in figure 1.

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Making contact with related work, we point out that the approach to remove basis-dependence by replacing the reference state with the (normalized) swap operator also builds upon an identification that was introduced for the special case of qubits in [27]. Furthermore, in terms of the conditional state formalism of [15], it is interesting to observe that the channel operator M_H resulting from the homocorrelation map applied to a CPTP map $\mathcal{M}$ is precisely the causal joint state obtained by conditioning the causal conditional state of $\mathcal{M}$ on the input $\mathbb{I}/{{d}_{A}}$. Thus, the homocorrelation map may also be seen as the composition of the Jamiołkowski isomorphism with the conditioning of the completely mixed state: $\mathcal{H}(\mathcal{M})=\sqrt{{{\mathbb{I}}_{A}}/{{d}_{A}}}\;\mathcal{J}(\mathcal{M})\sqrt{{{\mathbb{I}}_{A}}/{{d}_{A}}}$.

Example. A simple example may illustrate the operational relevance of $\mathcal{H}$. Consider the one-parameter family of qudit depolarizing channels [28], defined as

$$\begin{eqnarray*}&&{{\mathcal{D}}_{\eta }}(\rho )=(1-\eta )\;{\rm tr}(\rho )\frac{\mathbb{I}}{d}+\eta \rho .\end{eqnarray*}$$

The action of this channel commutes with all unitary channels in that ${{\mathcal{D}}_{\eta }}(U\rho {{U}^{\dagger }})=U{{\mathcal{D}}_{\eta }}(\rho ){{U}^{\dagger }}$. Under the homocorrelation map, the depolarizing channels are taken to operators with $U\otimes U$ symmetry, namely,

$$\begin{eqnarray}&&\mathcal{H}({{\mathcal{D}}_{\eta }})=(1-\eta )\frac{\mathbb{I}\otimes \mathbb{I}}{{{d}^{2}}}+\eta \frac{V}{d}.\end{eqnarray} \tag{ 7 }$$

Positive semi-definite operators of this form are the well-known Werner states [29]. Imagine that an observer does not know a priori whether her two measurements are made on distinct subsystems in a Werner state or on the same system before and after a depolarizing channel has been applied. If presented with a Werner state or depolarizing channel having $\eta \in [-\frac{1}{{{d}^{2}}-1},\frac{1}{d+1}]$ (see also section 3.1 for the meaning of this parameter range), the observer will not be able to distinguish between the two cases. The homocorrelation map makes this operational identification explicit. To contrast, the Choi map takes the depolarizing channels to so-called isotropic states [30],

$$\begin{eqnarray}&&\mathcal{C}({{\mathcal{D}}_{\eta }})=(1-\eta )\frac{\mathbb{I}\otimes \mathbb{I}}{{{d}^{2}}}+\eta \mid {{\Phi }^{+}}\rangle \langle {{\Phi }^{+}}\mid ,\end{eqnarray} \tag{ 8 }$$

where as before $|{{\Phi }^{+}}\rangle$ is the maximally entangled state. The isotropic states are defined by their symmetry with respect to $U\otimes {{U}^{*}}$ transformations. An observer in the scenario above would certainly be able to distinguish between the correlations of the depolarizing channel and the isotropic states, as long as $\eta \ne 0$.

The distinction between the maps $\mathcal{C}$ and $\mathcal{H}$ may be further appreciated by contrasting the sets of operators they produce. The set of CP maps forms a cone in the space of superoperators $\mathcal{L}({{\mathcal{H}}_{A}},{{\mathcal{H}}_{B}})$. Both $\mathcal{C}$ and $\mathcal{H}$ are cone-preserving maps (by linearity) from $\mathcal{L}({{\mathcal{H}}_{A}},{{\mathcal{H}}_{B}})$ to $\mathcal{B}({{\mathcal{H}}_{A}}\otimes {{\mathcal{H}}_{B}})$. While in the case of $\mathcal{C}$ the resulting cone is exactly the cone of bipartite states, in the case of $\mathcal{H}$ the corresponding cone is distinct from the cone spanned by states. One of our main findings here is that the correlations exhibited by bipartite states and the ones exhibited by quantum channels need not be equivalent. Furthermore, we find that this difference plays a role in their distinct joinability properties. The homocorrelation representation of channels provides us with a natural framework for exploring this difference: a channel and a state with differing correlations will be represented as distinct operators in the same operator space. These notions and their use in joinability are fleshed out in the remainder of this section.

The cone of positive operators plays a central role in defining joinability of quantum states. Analogously, the cone of homocorrelation-mapped channels will play a central role in defining joinability of quantum channels. These two cones are commonly known as positive-semidefinite operators (PSD) and positive-partial transpose operators (PPT), respectively. Here, we adopt an alternative terminology that emphasizes their physical significance, and refer to these cones and the operators they contain as state-positive and channel-positive, respectively. Formally:

Definition 2.4. (State-positivity (PSD)) An operator $M\in \mathcal{B}(\mathcal{H})$ is state-positive if ${\rm tr}(MP)\geqslant 0$ for all Hermitian projectors $P={{P}^{\dagger }}={{P}^{2}}\in \mathcal{B}(\mathcal{H})$. We notate this condition as $M{{\geqslant }_{{\rm st}}}0$ and emphasize that the resulting set is a self-dual cone.

Recall that a map $\mathcal{M}$ is a quantum channel if ${\rm tr}[\mathcal{C}(\mathcal{M})P]\geqslant 0$ for all $P={{P}^{\dagger }}={{P}^{2}}\in \mathcal{B}({{\mathcal{H}}_{A}}\otimes {{\mathcal{H}}_{B}})$ [21]. Using the relationships of figure 1, we translate this condition to one on the homocorrelation-mapped operator ${{M}_{H}}\equiv M=\mathcal{H}(\mathcal{M})$. Specifically, we define:

Definition 2.5. (Channel-positivity (PPT)) An operator $M\in \mathcal{B}({{\mathcal{H}}_{A}}\otimes {{\mathcal{H}}_{B}})$ is channel-positive with respect to the A–B bipartition if ${\rm tr}(M{{P}^{{{T}_{A}}}})\geqslant 0$ for all Hermitian projectors $P={{P}^{\dagger }}={{P}^{2}}\in \mathcal{B}({{\mathcal{H}}_{A}}\otimes {{\mathcal{H}}_{B}})$. We notate this condition $M{{\geqslant }_{{\rm ch}}}0$, and emphasize that the resulting set is, again, a self-dual cone.

A characterization of the intersection of the state- and channel-positive cones reveals a connection between quantum kinematics and dynamics.

Proposition 2.6. If a bipartite state $\rho \in \mathcal{B}({{\mathcal{H}}_{A}}\otimes {{\mathcal{H}}_{B}})$ and a quantum channel $\mathcal{M}\;:\mathcal{B}({{\mathcal{H}}_{A}})\to \mathcal{B}({{\mathcal{H}}_{B}})$ exhibit the same correlations (in the sense of definition 2.2), then ρ is PPT and $\mathcal{H}(\mathcal{M})$ is PSD.

Proof. Assume ρ and $\mathcal{M}$ exhibit the same correlations. Then, by proposition 2.3, $\rho =\mathcal{H}(\mathcal{M})$. Then $\mathcal{H}(\mathcal{M})$ is PSD because ρ is. The analogous statement for PPT follows from $\mathcal{H}(\mathcal{M})$ being PPT, but we find it instructive to give this calculation explicitly. Since $\mathcal{C}$ is related to $\mathcal{H}$ by a partial trace, we also have $\mathcal{H}{{(\mathcal{M})}^{{{T}_{A}}}}=\mathcal{C}(\mathcal{M})/{{d}_{A}}$. By the positivity preservation of $\mathcal{C}$, $\mathcal{M}$ being CPTP implies that $\mathcal{C}(\mathcal{M})$ is a positive operator. Thus, we have that ${{\rho }^{{{T}_{A}}}}=\mathcal{C}(\mathcal{M})/{{d}_{A}}$ is positive, making ρ PPT as claimed.

The above result may be used to directly connect statistical correlations in quantum states and channels to separability and, respectively, entanglement-breaking properties [32] of the channel:

Corollary 2.7. If the correlations of a bipartite state $\rho \in \mathcal{B}({{\mathcal{H}}_{A}}\otimes {{\mathcal{H}}_{B}})$ cannot be exhibited by a quantum channel, then the state is not separable. Similarly, if the correlations of a quantum channel $\mathcal{M}\;:\mathcal{B}({{\mathcal{H}}_{A}})\to \mathcal{B}({{\mathcal{H}}_{B}})$ cannot be exhibited by a bipartite state, then the channel is not entanglement breaking.

Proof. Since the correlations of ρ cannot be exhibited by a quantum channel, the operator is not PPT, by proposition 2.6. Then, by the Peres–Horodecki criterion [31], the state is necessarily entangled. For the second part, since the correlations of the channel cannot be exhibited by a bipartite state, the homocorrelation mapped channel $\mathcal{H}(\mathcal{M})$ is not PSD, by proposition 2.6. Recall that an entanglement breaking channel $\mathcal{E}:\mathcal{B}({{\mathcal{H}}_{A}})\to \mathcal{B}({{\mathcal{H}}_{B}})$ can always be written in the 'Holevo form' [32, 33],

$$\begin{eqnarray*}&&\mathcal{E}(\rho )=\mathop{\sum }\limits_{k}{{\tau }_{k}}{\rm tr}({{\sigma }_{k}}\rho ),\end{eqnarray*}$$

where ${{\tau }_{k}}\in \mathcal{B}({{\mathcal{H}}_{B}})$ and ${{\sigma }_{k}}\in \mathcal{B}({{\mathcal{H}}_{A}})$ are PSD operators normalized to ensure that $\mathcal{E}$ is TP. If $\mathcal{M}$ were entanglement breaking, then by using the definition of the Jamiołkowski isomorphism in equation (2), $\mathcal{H}(\mathcal{M})$ would have the form

$$\begin{eqnarray*}\begin{array}{rcl} \mathcal{H}(\mathcal{M}) & = & \frac{1}{{{d}_{A}}}\mathop{\sum }\limits_{k}\mathop{\sum }\limits_{ij}\;|i{{\rangle }_{1}}\langle j|\otimes {{\tau }_{k}}{\rm t}{{{\rm r}}_{2}}{{({{\sigma }_{k}}|j\rangle }_{2}}\langle i|) \\ {} & = & \frac{1}{{{d}_{A}}}\mathop{\sum }\limits_{k}{\rm t}{{{\rm r}}_{2}}[({{\mathbb{I}}_{1}}\otimes {{\sigma }_{k}})V]\otimes {{\tau }_{k}}=\frac{1}{{{d}_{A}}}\mathop{\sum }\limits_{k}\;{{\sigma }_{k}}\otimes {{\tau }_{k}}, \\ \end{array}\end{eqnarray*}$$

where the subscripts 1 (2) refer to the first (second) copy in ${{\mathcal{H}}_{A}}\otimes {{\mathcal{H}}_{A}}$ respectively. Thus, $\mathcal{H}(\mathcal{M})$ is a (separable) bipartite quantum state. Since $\mathcal{H}(\mathcal{M})$ is not PSD, and hence, not a quantum state, $\mathcal{M}$ is not entanglement breaking.

A pictorial representation of the geometry of state- and channel-positive cones is shown in figure 2 for two-qudit Werner operators as in equation (7). This will aid the understanding of agreement bounds for quantum states versus channels (see section 4). It is also worth noting that recent work has also emphasized the difference between the set of state-positive (positive) and channel-positive (PPT) operators [34], in the context of showing the extent to which local measurements on quantum systems may be used to distinguish between causal-influence and common-cause relationships. Loosely speaking, statistics consistent with a non-PPT operator indicate some degree of common cause, while those consistent with a non-positive operator indicate some degree of causal influence—hence implying that entanglement also provides a quantum advantage for causal inference as compared to the analogous classical setting.

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We are now poised to use the homocorrelation representation to formulate joinability problems for quantum channels. The channel-positive operators provide an alternative set with which to define the allowed joining operators W. As a warm-up, we rephrase the channel-joinability problem that was posed in the Introduction. Consider quantum channels from ${{\mathcal{H}}_{A}}$ to ${{\mathcal{H}}_{B}}\otimes {{\mathcal{H}}_{C}}$. Under the homocorrelation map, these correspond to tripartite operators lying in the channel-positive cone, notated ${{W}_{A|BC}}$. The partial traces ${\rm t}{{{\rm r}}_{C}}$ and ${\rm t}{{{\rm r}}_{B}}$ take channel-positive operators in ${{W}_{A|BC}}$ to channel-positive operators in ${{W}_{A|B}}$ and ${{W}_{A|C}}$, respectively; that is, operators in ${{W}_{A|B}}$ and ${{W}_{A|C}}$ correspond to valid quantum channels via the homocorrelation map. The corresponding channel-joining scenario is then defined as $({{W}_{A|BC}},\{{\rm t}{{{\rm r}}_{C}},{\rm t}{{{\rm r}}_{B}}\})$. A channel-joinability (or extension) problem presents two channel operators ${{M}_{AB}}\in {{W}_{A|B}}$ and ${{M}_{AC}}\in {{W}_{A|C}}$ and seeks to determine the existence of a channel operator ${{M}_{ABC}}\in {{W}_{A|BC}}$ which reduces to the two channel operators in question. In general, we thus have the following:

Definition 2.8. (Channel-joinability) Given a joinability scenario described by the pair $(W{{\geqslant }_{{\rm ch}}}0\},\{{\rm t}{{{\rm r}}_{{\hat{\ell }}}}\})$, the reduced operators $\{{{M}_{\ell }}\}\in \{{{R}_{\ell }}\}$ are joinable if there exists a joint operator $M\in W$ such that ${\rm t}{{{\rm r}}_{{\hat{\ell }}}}(M)={{M}_{\ell }}$ for all ℓ.

A channel joinability problem can in principle be stated using the Choi instead of the homocorrelation map, as done in [16]. However, as we argued, the latter provides a platform to directly compare the joinability of states and channels of equivalent correlations. For instance, it will allow us to simultaneously compare the joinability of local-unitary-invariant states and channels, and consequently to compare these both to the joinability of analogous classical probability distributions (cf figure 5).

Before proceeding to the general notion of joinability, we remark that allowed joining operators in W have thus far been considered to be either state-positive or channel-positive. However, from a mathematical standpoint, a viable joinability problem only needs W to be a convex cone. To investigate this generalization and to further meld state and channel joining, we consider a third type of positivity that we call local-positivity. This notion is equivalent to both block-positivity and to map-positivity (not necessarily CP) [35, 36], in that by representing linear maps using $\mathcal{H}$, the cone of (transformed) positive maps is equal to the cone of bipartite block-positive operators. In fact, it was precisely the correspondence between block-positive operators and positive maps that led Jamiołkowski to use this isomorphism in [20]. Formally:

Definition 2.9. (Local-positivity) An operator $M\in \mathcal{B}({{\mathcal{H}}_{A}}\otimes {{\mathcal{H}}_{B}})$ is local-positive with respect to the A–B factorization if ${\rm tr}(M{{P}_{A}}\otimes {{P}_{B}})\geqslant 0$ for all pure states ${{P}_{A}}=P_{A}^{2}\in \mathcal{B}({{\mathcal{H}}_{A}})$ and ${{P}_{B}}=P_{B}^{2}\in \mathcal{B}({{\mathcal{H}}_{B}})$. We notate this condition $M{{\geqslant }_{{\rm loc}}}0$.

Local-positivity is readily generalized to more than two systems. The set of channel-positive operators and state-positive operators are each sub-cones of the local-positive operators, as local-positivity clearly is a weaker condition. In particular, joinability with respect to the state-positive or channel-positive cone implies joinability with respect to the local-positive cone. Local-positive operators are directly relevant to QIP, in particular because they may serve as entanglement witnesses [37].

Compared to state- and channel-positivity, the notion of local-positivity yields a less-strict definition of quantum joinability, which allows for closer comparison to joinability limitations derived from classical probability. We take a brief detour to explain how classical probability theory limits quantum joinability. Consider three bipartite quantum states ${{\rho }_{AB}}$, ${{\rho }_{AC}},$ and ${{\rho }_{BC}}$. By choosing a product basis for the three systems $\{|{{i}_{A}}{{j}_{B}}{{k}_{C}}\rangle \}$, we may correspond each density operator to a probability distribution via ${{\rho }_{AB}}\mapsto {\rm p}({{i}_{A}}{{j}_{B}})\equiv \langle {{i}_{A}}{{j}_{B}}|{{\rho }_{AB}}|{{i}_{A}}{{j}_{B}}\rangle$, and so on. Assume the three density operators could be joined by the state w_ABC. Then, to w_ABC, we could associate a joint probability distribution ${\rm p}({{i}_{A}}{{j}_{B}}{{k}_{C}})$ whose marginals would necessarily be ${\rm p}({{i}_{A}}{{j}_{B}})$, etc. In other words, if ${{\rho }_{AB}}$, ${{\rho }_{AC}}$, and ${{\rho }_{BC}}$ were joinable, then their corresponding classical distributions would be joinable. The contrapositive of this statement describes how classical probability limits quantum joinability: if ever the ${\rm p}({{i}_{A}}{{j}_{B}})$, etc are not joinable with one another, then the corresponding density operators ${{\rho }_{AB}}$, etc are not be joinable with one another. Thus, classical probability theory provides necessary conditions for quantum joinability. This reasoning holds for channel-joinability scenarios and local-positive joinability scenarios alike: the positivity of the derived classical distributions (e.g. ${\rm p}({{i}_{A}}{{j}_{B}}{{k}_{C}})$) is ensured by channel-positivity and by local-positivity. In general, given a choice of local bases $\{|{{i}_{A}}{{j}_{B}}{{k}_{C}}\ldots {{m}_{E}}\rangle \}$, the joining operator is constrained by

$$\begin{eqnarray*}&&{\rm p}({{i}_{A}}{{j}_{B}}{{k}_{C}}\ldots {{m}_{E}})\equiv {\rm tr}({{w}_{ABC\ldots Z}}|{{i}_{A}}\rangle \langle {{i}_{A}}|\otimes \ldots |{{m}_{E}}\rangle \langle {{m}_{E}}|)\geqslant 0.\end{eqnarray*}$$

Local-positivity is equivalent to requiring the above to hold with respect to all choices of basis. In order to distinguish between local-positive constraints and those of classical probability theory we consider only classical probability distributions derived from local bases that are identified with one another ($\{|i\rangle \}\equiv \{|{{i}_{A}}\rangle \}=\{|{{i}_{B}}\rangle \}=\ldots$). Then, local-positivity constraints are more general (hence stricter) than classical probability constraints in that the local projectors of the former may be oblique (i.e. neither orthogonal nor parallel) with respect to one another. In section 3.2 we show how Werner operators provide an example of this strict inclusion.

Another way of viewing the various definitions of positivity is to understand the subscript on the inequality to indicate the dual cone from which inner products with M must be positive. For $M{{\geqslant }_{{\rm st}}}0$, $M{{\geqslant }_{{\rm ch}}}0$, and $M{{\geqslant }_{{\rm loc}}}0$, the respective dual cones are the positive span of rank-one projectors, the positive span of partially-transposed projectors, and the positive span of product projectors (from which the trace-one condition confines to the set of separable states). We note that the first two cones are self-dual (and, furthermore, symmetric [38]), while the local-positive cone is not. With several relevant examples of positivity established, each being a different convex set with which to define W, we are in a position to finally give the following:

Definition 2.10. (General quantum joinability) Let W be a convex cone in $\mathcal{B}({{\mathcal{H}}^{(N)}})$, and $\{{\rm t}{{{\rm r}}_{{\hat{\ell }}}}\}$ be partial traces with $\ell \subset {{\mathbb{Z}}_{N}}$. Given the joinability scenario $(W,\{{\rm t}{{{\rm r}}_{{\hat{\ell }}}}\})$, the operators $\{{{M}_{\ell }}\}\in \{{{R}_{\ell }}\}$ are joinable if there exists a joining operator $w\in W$ such that ${\rm t}{{{\rm r}}_{{\hat{\ell }}}}(w)={{M}_{\ell }}$ for all ℓ.

This general definition encompasses the various joinability problems referenced in the Introduction. Specifically, if W is the set of quantum states on a multipartite system, the joinability problem reduces to the quantum marginal problem, whereas if W consists of channel-positive operators describing quantum channels from one multipartite system to another, one recovers the channel-joining problem instead. Specific instances of this problem are the optimal asymmetric cloning problem [17, 18, 39], the symmetric cloning problem [40, 41], and the k-extendibility problem for quantum maps [42]. In addition to providing a unified perspective, our approach has the important advantage that different classes of joinability problems may be mapped into one another, in such a way that a solution to one provides a solution to another. This is made formal in the following:

Proposition 2.11. Let W and $W^{\prime}$ be two cones of operators acting on the space ${{\mathcal{H}}^{(N)}}$, $\{{\rm t}{{{\rm r}}_{{\hat{\ell }}}}\}$ a set of partial traces that apply to both cones, and $\phi :\mathcal{B}({{\mathcal{H}}^{(N)}})\to \mathcal{B}({{\mathcal{H}}^{(N)}})$ a linear cone-preserving map (i.e. $\phi (W)\subseteq W^{\prime}$), which permits reduced actions ${{\phi }_{\ell }}$ satisfying ${{\phi }_{\ell }}\circ {\rm t}{{{\rm r}}_{{\hat{\ell }}}}={\rm t}{{{\rm r}}_{{\hat{\ell }}}}\;\circ \phi$. If $\{{{M}_{\ell }}\}\in \{{\rm t}{{{\rm r}}_{{\hat{\ell }}}}(W)\}$ is joinable with respect to W, then $\{{{\phi }_{\ell }}({{M}_{\ell }})\}\in \{{\rm t}{{{\rm r}}_{{\hat{\ell }}}}(W^{\prime} )\}$ is joinable with respect to $W^{\prime}$.

Proof. Assume that w is a valid joining operator for the operators $\{{{M}_{\ell }}\}\in \{{\rm t}{{{\rm r}}_{{\hat{\ell }}}}(W)\}$. Then, the set of operators $\{{{\phi }_{\ell }}({{M}_{\ell }})\}\in \{{\rm t}{{{\rm r}}_{{\hat{\ell }}}}(W^{\prime} )\}$ is joined by the operator $\phi (w),$ since ${\rm t}{{{\rm r}}_{{\hat{\ell }}}}[\phi (w)]={{\phi }_{\ell }}({\rm t}{{{\rm r}}_{{\hat{\ell }}}}[w])={{\phi }_{\ell }}({{M}_{\ell }})$ and $\phi (w)\in W^{\prime}$.

This is shown in the commutative diagram of figure 3. In what follows, a stronger corollary of the above result will be employed:

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Corollary 2.12. Let $\phi :\mathcal{B}({{\mathcal{H}}^{(N)}})\to \mathcal{B}({{\mathcal{H}}^{(N)}})$ be an invertible linear map satisfying $\phi (W)=W^{\prime}$, with invertible reduced actions ${{\phi }_{\ell }}$ satisfying ${{\phi }_{\ell }}\circ {\rm t}{{{\rm r}}_{{\hat{\ell }}}}={\rm t}{{{\rm r}}_{{\hat{\ell }}}}\;\circ \phi$ (and similarly for their inverses). Then a set of operators $\{{{M}_{\ell }}\}\in \{{\rm t}{{{\rm r}}_{{\hat{\ell }}}}(W)\}$ is joinable if and only if the set of operators $\{{{\phi }_{\ell }}({{M}_{\ell }})\}$ is joinable.

Proof. The forward implication follows from proposition 2.11, while the backwards implication follows from the fact that ϕ and the ${{\phi }_{\ell }}$ are invertible, along with the contrapositive of proposition 2.11.

The joinability-problem isomorphism we make use of is the partial transpose map. This transformation permits a natural reduced action, namely, partial transpose on the subsystems that remain after partial trace (see also figure 3). As explained, the partial transpose is a positivity-preserving bijection between states and channel operators (up to normalization). Thus, by determining the joinable–unjoinable demarcation for a class of states, we will also determine the joinable–unjoinable demarcation for a corresponding class of channel-operators.

Bipartite Werner operators obey collective unitary invariance. By standard Shur–Weyl duality [43], these operators have the form given in equation (7), which for convenience we repeat here,

$$\begin{eqnarray}&&\rho (\eta )=(1-\eta )\frac{\mathbb{I}}{{{d}^{2}}}+\eta \frac{V}{d},\end{eqnarray} \tag{ 13 }$$

with V being the swap operator defined earlier. The $\pm 1$ eigenvalues of V dictate the state positive range of $\rho (\eta )$ to be $-\frac{1}{d-1}\leqslant \eta \leqslant \frac{1}{d+1}$, corresponding to the qudit Werner states we already mentioned. This parameterization is chosen so that η is a 'correlation' measure: if d = 2, then $\eta =-1$ corresponds to the singlet state, $\eta =0$ to the maximally mixed state. The correlation parameter is related to the 'degree of agreement', discussed above, via

$$\begin{eqnarray}&&\eta =\frac{d}{d-1}\left( \alpha -\frac{1}{d} \right).\quad \end{eqnarray} \tag{ 14 }$$

We choose to present our results in terms of η because it simplifies the expressions. Note that the value $\eta =1$ ($\alpha =1$) does not correspond to a valid quantum state, but rather to a valid quantum channel operator that expresses perfect correlation for all possible collective measurements. As seen in the example of section 2.1, channel-positive operators with $U\otimes U$-invariance correspond to depolarizing channels, with $\eta =1$ labeling the identity channel. It is known that channel-positivity of the depolarizing map is ensured provided that $-\frac{1}{{{d}^{2}}-1}\leqslant \eta \leqslant 1$ [44]. We find it instructive to re-stablish state- and channel-positivity bounds using the Choi isomorphism.

To this end, we enlarge the above class of Werner operators to include the operators which are obtained from partial transposition of Werner operators. From equation (8), such operators are the ${{U}^{*}}\otimes U$-invariant isotropic operators. The span of operators exhibiting either $U\otimes U$ or ${{U}^{*}}\otimes U$ invariance will be invariant under the action of collective orthogonal transformations, that is operators that may be written as $O\otimes O\equiv U\otimes U={{U}^{*}}\otimes U$ for some unitary U ³ . This extended class of operators provides a useful test bed in that it expresses familiar states as well as channel operators of well-known quantum channels, while being non-trivially closed under partial transpose. A general operator in this space may be parameterized as follows [46]:

$$\begin{eqnarray}&&\rho (\eta ,\beta )=(1-\eta -\beta )\frac{\mathbb{I}}{{{d}^{2}}}+\eta \frac{V}{d}+\beta \frac{{{V}^{{{T}_{A}}}}}{d}.\end{eqnarray} \tag{ 15 }$$

In particular, the operator $\rho (0,1)$ is a generic Bell state on two qudits, $\rho (0,-1/(d-1))$ is the maximally entangled Werner state (e.g., the singlet state for d = 2), $\rho (1,0)$ is the identity channel, and $\rho (0,0)$ is the completely mixed state (or the completely depolarizing channel). We can then establish the following:

Proposition 3.1. A bipartite operator $\rho (\eta )$ with collective unitary invariance is channel-positive if and only if $-\frac{1}{{{d}^{2}}-1}\leqslant \eta \leqslant 1$.

Proof. By definition, an operator is channel-positive if it is PPT. Furthermore, the Brauer operators in equation (15) satisfy the property $\rho {{(\eta ,\beta )}^{{{T}_{A}}}}=\rho (\beta ,\eta )$. Thus, we need only determine the state-positivity of the operators $\rho (\beta ,\eta )$ to determine the bounds which channel-positivity places on η and β. The eigenspaces of any Brauer operator are the anti-symmetric subspace, the one-dimensional space spanned by $|{{\Phi }^{+}}\rangle$, and the space spanned by vectors $|y\rangle$ satisfying ${{\langle y|(|y\rangle )}^{*}}=0$ ⁴ , for which we label the projectors P_A, ${{P}_{+}}$, and P_Y, respectively. For the operator $\rho (\beta ,\eta )$, where η and β have been switched by the partial transpose, the eigenvalues are as follows:

$$\begin{eqnarray*}\begin{array}{rcl} \rho (\beta ,\eta ){{P}_{A}} & = & [(1-\eta -\beta )/{{d}^{2}}-\beta /d]{{P}_{A}}, \\ \rho (\beta ,\eta ){{P}_{+}} & = & [(1-\eta -\beta )/{{d}^{2}}+\beta /d+\eta ]{{P}_{+}}, \\ \rho (\beta ,\eta ){{P}_{Y}} & = & [(1-\eta -\beta )/{{d}^{2}}+\beta /d]{{P}_{Y}}. \\ \end{array}\end{eqnarray*}$$

Hence, channel-positivity of the bipartite Brauer operators is ensured by

$$\begin{eqnarray*}&&1\geqslant (d+1)\beta +\eta ,\quad 1\geqslant -(d-1)\beta -({{d}^{2}}-1)\eta ,\quad 1\geqslant -(d-1)\beta +\eta .\end{eqnarray*}$$

In particular, we recover that the channel-positive range for $U\otimes U$-invariant operators ($\beta =0$) is $-\frac{1}{{{d}^{2}}-1}\leqslant \eta \leqslant 1$, whereas the state-positive range (obtained by swapping η and β in the above inequalities) is $-\frac{1}{d-1}\leqslant \eta \leqslant \frac{1}{d+1}$.

Reasoning in a similar manner, we can also obtain the ranges of local-positivity:

Proposition 3.2. A bipartite operator $\rho (\eta )$ with collective unitary invariance is local-positive if and only if $-\frac{1}{d-1}\leqslant \eta \leqslant 1$.

Proof. Local positivity is ensured by the non-negativity of expectation values with respect to product vectors, say, $\langle \phi \psi |\rho (\eta ,\beta )|\phi \psi \rangle$. Explicitly, we have

$$\begin{eqnarray*}&&\frac{1-\eta -\beta }{{{d}^{2}}}+\frac{\eta }{d}|\langle \phi |\psi \rangle {{|}^{2}}+\frac{\beta }{d}|{{\langle \phi |(|\psi \rangle )}^{*}}\mid {{}^{2}}\geqslant 0,\end{eqnarray*}$$

with $0\leqslant |\langle \phi |\psi \rangle {{|}^{2}},|{{\langle \phi |(|\psi \rangle )}^{*}}{{|}^{2}}\leqslant 1$. We now show that all four extremal value pairings of these factors may be achieved, leading to four inequalities whose satisfaction guarantees that of all others. Consider $|x\rangle$ and $|y\rangle$ satisfying $|x{{\rangle }^{*}}=|x\rangle$ and $|y{{\rangle }^{*}}=|\bar{y}\rangle$, where the bar indicates a vector orthogonal to the original vector. Setting $|\phi \psi \rangle =|x\bar{x}\rangle$ achieves $|\langle \phi |\psi \rangle {{|}^{2}}=0$, $|{{\langle \phi |(|\psi \rangle )}^{*}}{{|}^{2}}=0$; setting $|\phi \psi \rangle =|y\bar{y}\rangle$ achieves $|\langle \phi |\psi \rangle {{|}^{2}}=0$, $|{{\langle \phi |(|\psi \rangle )}^{*}}{{|}^{2}}=1$; setting $|\phi \psi \rangle =|yy\rangle$ achieves $|\langle \phi |\psi \rangle {{|}^{2}}=1$, $|{{\langle \phi |(|\psi \rangle )}^{*}}{{|}^{2}}=0$; and setting $|\phi \psi \rangle =|xx\rangle$ achieves $|\langle \phi |\psi \rangle {{|}^{2}}=1$, $|{{\langle \phi |(|\psi \rangle )}^{*}}{{|}^{2}}=1$. Thus, the following four extremal inequalities bound the local-positive region of the η–β space:

$$\begin{eqnarray*}\begin{array}{rcl} 0\leqslant {{\langle \rho (\eta ,\beta )\rangle }_{xx}} & = & (1-\eta -\beta )/{{d}^{2}}+\eta /d+\beta /d, \\ 0\leqslant {{\langle \rho (\eta ,\beta )\rangle }_{x\bar{x}}} & = & (1-\eta -\beta )/{{d}^{2}}, \\ 0\leqslant {{\langle \rho (\eta ,\beta )\rangle }_{yy}} & = & (1-\eta -\beta )/{{d}^{2}}+\eta /d, \\ 0\leqslant {{\langle \rho (\eta ,\beta )\rangle }_{y\bar{y}}} & = & (1-\eta -\beta )/{{d}^{2}}+\beta /d. \\ \end{array}\end{eqnarray*}$$

More compactly, we may write

$$\begin{eqnarray*}&&-\frac{1}{d-1}\leqslant \eta +\beta \leqslant 1,\quad -(d-1)\eta +\beta \leqslant 1,\quad \eta -(d-1)\beta \leqslant 1,\end{eqnarray*}$$

whereby the desired result follows⁵ .

A pictorial summary of the three positivity bounds provided by propositions 3.1 and 3.2 is presented in figure 4.

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Having characterized all types of positivity for the (bipartite) operators to be joined, we now turn to characterize the positivity for the (tripartite) joining operators. For each positive tripartite set ($W{{\geqslant }_{{\rm st}}}0$, $W{{\geqslant }_{{\rm ch}}}0$, and $W{{\geqslant }_{{\rm loc}}}0$), we obtain the trios of joinable bipartite operators by simply applying the three partial traces $({\rm T}{{{\rm r}}_{A}},{\rm T}{{{\rm r}}_{B}},{\rm T}{{{\rm r}}_{C}})$ to each positive operator. Our approach is to obtain an expression for the positivity boundary of the tripartite operators in terms of operator space coordinates, and then re-express this boundary in terms of parameters for the reduced operators (the three Werner parameters in this case). It is important to realize that the Werner symmetry, satisfied by the bipartite reduced operators, can be exploited to narrow down the set of joining operators to consider. Namely, if a trio of bipartite Werner operators is joined by some w with given positivity properties, then the trio is also joined by a 'twirled' operator $\tilde{w}$ with the same positivity properties, given by (cf equation (13) in [6]):

$$\begin{eqnarray*}&&\tilde{w}=\int {\rm d}\mu (U)U\otimes U\otimes Uw{{(U\otimes U\otimes U)}^{\dagger }}.\end{eqnarray*}$$

Thus, a trio of bipartite Werner operators is joinable if and only if it is joinable by some tripartite Werner operator. For state- and channel-positivity, the desired characterization follows directly from the analysis reported in our previous work [6].

Corollary 3.3. With reference to the parameterization of equation (13), we have that:

• (i)  
  Three qudit Werner states with parameters $({{\eta }_{AB}},{{\eta }_{AC}},{{\eta }_{BC}})$ are joinable relative to the $({{W}_{ABC}}{{\geqslant }_{{\rm st}}}0,\{{\rm t}{{{\rm r}}_{A}},{\rm t}{{{\rm r}}_{B}},{\rm t}{{{\rm r}}_{C}}\})$ scenario if and only if

  $$\begin{eqnarray*}&&\left\{ \begin{array}{ccccccccccccccc} \frac{1}{2}(1-{{\eta }_{AB}}-{{\eta }_{AC}}-{{\eta }_{BC}})\geqslant |{{\eta }_{AB}}+\omega {{\eta }_{AC}}+{{\omega }^{2}}{{\eta }_{BC}}|,\quad \omega \equiv {{{\rm e}}^{{\rm i}2\pi /3}}, \\ {{\eta }_{AB}}+{{\eta }_{AC}}+{{\eta }_{BC}}\geqslant -1, \\ \end{array} \right.\end{eqnarray*}$$

  for d = 2, while for $d\geqslant 3$ they need only satisfy

  $$\begin{eqnarray*}&&\frac{3}{2d(d\mp 1)}\left( 1\pm (d\mp 1)({{\eta }_{AB}}+{{\eta }_{AC}}+{{\eta }_{BC}}) \right)\geqslant |{{\eta }_{AB}}+\omega {{\eta }_{AC}}+{{\omega }^{2}}{{\eta }_{BC}}|.\end{eqnarray*}$$
• (ii)  
  Three qudit channel-positive Werner operators with parameters $({{\eta }_{AB}},{{\eta }_{AC}},{{\eta }_{BC}})$ are channel-joinable relative to the $({{W}_{A|BC}}{{\geqslant }_{{\rm ch}}}0,\{{\rm t}{{{\rm r}}_{A}},{\rm t}{{{\rm r}}_{B}},{\rm t}{{{\rm r}}_{C}}\})$ scenario if and only if

  $$\begin{eqnarray*}&&\begin{array}{l} \quad \frac{1}{d-1}+{{\eta }_{AB}}+{{\eta }_{AC}}-{{\eta }_{BC}}\geqslant |\frac{2}{d-1}+d{{\eta }_{BC}}+\sqrt{\frac{2d}{d-1}}({{{\rm e}}^{{\rm i}\theta }}{{\eta }_{AB}}+{{{\rm e}}^{-{\rm i}\theta }}{{\eta }_{AC}})|, \\ \quad {{{\rm e}}^{{\rm i}\theta }}\equiv \sqrt{(d-1)/2d}\pm {\rm i}\sqrt{(d+1)/2d}. \\ \end{array}\end{eqnarray*}$$

  The channel-joinability limitations in the other two scenarios $B|AC$ and $C|AB$ may be obtained by permuting the ηs accordingly.

Proof. Result (i) above corresponds to theorem 3 in [6], re-expressed in terms of the parametrization of equation (13) (with reference to the notation of equations (15)–(17) in [6], one has ${{\eta }_{\ell }}=(d/({{d}^{2}}-1))(\Psi _{\ell }^{-}-1/2)$, $\ell =AB,AC,BC)$.

In order to establish (ii), note that $\mathcal{C}$ may be used to translate any ${{U}^{*}}\otimes U$-invariant state-positive joinability problem into a $U\otimes U$-invariant channel-positive joinability problem, drawing on corollary 2.12. Explicitly, under $\mathcal{C}$ (partial transpose in the case of operators), the ${{U}^{*}}\otimes U\otimes U$-invariant state-positive operators ${{W}_{A*BC}}$ are in one-to-one correspondence with the $U\otimes U\otimes U$-invariant channel-positive operators ${{W}_{A|BC}}$. Hence, by the joinability isomorphism induced by the partial transpose, the solution to a joinability problem of the scenario $({{W}_{{{A}^{*}}BC}},\{{\rm t}{{{\rm r}}_{A}},{\rm t}{{{\rm r}}_{B}},{\rm t}{{{\rm r}}_{C}}\})$ gives a solution to a corresponding joinability problem of $({{W}_{A|BC}},\{{\rm t}{{{\rm r}}_{A}},{\rm t}{{{\rm r}}_{B}},{\rm t}{{{\rm r}}_{C}}\})$. Thus, to obtain the depolarizing channel joinability boundaries, we simply translate the isotropic state parameters of equations (20)–(21) in [6] into η parameters.

The joinability limitations of all four scenarios are depicted in figure 5. In the qubit case we are granted some liberty in perspective. In the space depicting the joinability limitations, the coordinates are truly the reduced-state Werner parameters; however, we may also view them as coordinates in operator space of the tripartite joining operator, with the axes corresponding to the orthogonal operator basis $\{\frac{1}{4}({{V}_{AB}}-\mathbb{I}/2),\frac{1}{4}({{V}_{AC}}-\mathbb{I}/2),\frac{1}{4}({{V}_{BC}}-\mathbb{I}/2)\}$. This justifies an abuse of terminology, identifying the space of reduced state coordinates with the space of relevant joining operators.

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By rewriting equations (9)–(12) in terms of the η-parametrization by using equation (14), we determine the tetrahedra given in figures 5(a)–(b). In the qubit case, it is intriguing that the inclusion of the quantum channel-joinability limitations allows one to regain the tetrahedral symmetry imposed by the classical limitations; whereas each scenario on its own expresses a continuous rotational symmetry⁶ that is not reflected classically. In other words, if we consider the joinability scenario defined by

$$\begin{eqnarray*}&&({\rm span}\{{{W}_{ABC}},{{W}_{A|BC}},{{W}_{B|AC}},{{W}_{C|AB}}\},\{{\rm t}{{{\rm r}}_{A}},{\rm t}{{{\rm r}}_{B}},{\rm t}{{{\rm r}}_{C}}\}),\end{eqnarray*}$$

the joinable bipartite operators respect the tetrahedral symmetry of the classical joinability bounds. This amounts to ask: What trios of bipartite correlations—as derivable from either quantum states or channels, or from probabilistic combinations of the two—may be obtained from the measurements on three systems? Though the result expresses the tetrahedral symmetry of the classical joinability limitations, these classical joinability limitations do not suffice to enforce the stricter quantum limitations, as manifest in the fact that the corners of the classical joinability tetrahedron are not reached by the quantum boundaries. We diagnose such limitations as strictly quantum features that do not have classical analogues—as we will discuss later in this work.

We now explore how local-positive joinability (a weaker restriction, as noted) relates to the state/channel-joinability limitations above, as well as to the underlying classical limitations. As of yet, we only know that the local-positive limitations will lie between the classical and the quantum boundaries. Since obtaining a simple analytical characterization for arbitrary subsystem dimension d appears challenging in the local-positive setting, and useful insight may already be gained in the lowest-dimensional (qubit) setting, we focus on d = 2 in this section. Our main result is the following:

Theorem 3.4. With reference to equation (13), three qubit local-positive Werner operators with parameters $({{\eta }_{AB}},{{\eta }_{AC}},{{\eta }_{BC}})$ are joinable by a local-positive tripartite operator if and only if the following conditions hold:

$$\begin{eqnarray}\begin{array}{ll} 1+{{\eta }_{AB}}+{{\eta }_{AC}}+{{\eta }_{BC}}\;\geqslant & 0,\quad 1+{{\eta }_{AB}}-{{\eta }_{AC}}-{{\eta }_{BC}}\geqslant 0, \\ 1-{{\eta }_{AB}}+{{\eta }_{AC}}-{{\eta }_{BC}}\;\geqslant & 0,\quad 1-{{\eta }_{AB}}-{{\eta }_{AC}}+{{\eta }_{BC}}\geqslant 0, \\ \end{array}\end{eqnarray} \tag{ 16 }$$

and

$$\begin{eqnarray*}&&2{{\eta }_{AB}}{{\eta }_{AC}}{{\eta }_{BC}}-\eta _{AB}^{2}\eta _{AC}^{2}-\eta _{AB}^{2}\eta _{BC}^{2}-\eta _{AC}^{2}\eta _{BC}^{2}\geqslant 0.\end{eqnarray*}$$

We note that the first four linear inequalities delimit the tetrahedral classical joinability boundary determined in [6], once expressed in terms of the η variables via equation (14). The proof is rather lengthy and deferred to a separate appendix. The resulting boundary is depicted in figure 6; this shape and its determining equation are recognized as the convex hull of the Roman surface (aka Steiner surface) [35, 50]. Comparing with the classical joinability limitations (tetrahedron), we see that, still, the quantum joinability limitations arising from from local-positivity are more strict. However, the Roman surface encloses the region of state/channel joinability from figure 5(a), as expected, since both state-positive and channel-positive imply local-positive. To shed light on the cause of the stricter quantum boundary (i.e., the local-positive joining boundary) here, we can explicitly construct a product-state projector, whose likelihood in a measurement would be negative if joinability outside of this shape were allowed. The family of joining states w that we need to consider (see appendix) may be parameterized in terms of the bipartite reduced state Werner parameters as

$$\begin{eqnarray*}&&w({{\eta }_{AB}},{{\eta }_{AC}},{{\eta }_{BC}})=\frac{1}{8}\mathbb{I}+\frac{{{\eta }_{AB}}}{4}({{V}_{AB}}-\mathbb{I}/2)+\frac{{{\eta }_{AC}}}{4}({{V}_{AC}}-\mathbb{I}/2)+\frac{{{\eta }_{BC}}}{4}({{V}_{BC}}-\mathbb{I}/2).\end{eqnarray*}$$

Consider, in particular, the following state on A–B–C:

$$\begin{eqnarray*}&&|\psi \rangle =\left[ \begin{array}{ccccccccccccccc} 1 \\ 0 \\ \end{array} \right]\otimes \left[ \begin{array}{ccccccccccccccc} {\rm cos} 2\pi /3 \\ {\rm sin} 2\pi /3 \\ \end{array} \right]\otimes \left[ \begin{array}{ccccccccccccccc} {\rm cos} 4\pi /3 \\ {\rm sin} 4\pi /3 \\ \end{array} \right],\end{eqnarray*}$$

which corresponds to the pure product state with the local Bloch vectors pointing 'as anti-parallel with one another as possible'. Computing its expectation with respect to $w({{\eta }_{AB}}={{\eta }_{AC}}={{\eta }_{BC}}\equiv \eta )$, the largest value of η that admits a non-negative value is $\eta =2/3$. Hence, local-positivity limits the joinability of these Werner operators to a maximum of $\eta =2/3$. The operational interpretation of this result deserves attention. Consider local projective measurements made on each of three qubit systems, and consider the three systems to have a collective unitary symmetry, in the sense that there are no preferred local bases. In our general picture, the systems need not be three distinct systems—they may also be systems at two different points in time. Then, local positivity enforces the rule that 'all probabilities arising from such measurements must be non-negative'. In the example above (i.e. ${{\eta }_{AB}}={{\eta }_{AC}}={{\eta }_{BC}}$), this implies that the three equal correlations (as measured by the ηs) can never exceed 2/3. The classical joinability constraints are derived from projectors which are parallel or orthogonal to one another. Thus, the 'relative obliqueness' of the three projectors in this example is what allows for the local-positive constraint to be stricter than classical constraints. Notwithstanding, both limitations express a tetrahedral symmetry, that is, symmetry with respect to individually inverting two axes. In contrast, the state-joining and channel-joining scenarios manifest a preference towards the negative axis (anticorrelation) and the positive axis (correlation) of the ηs, respectively.

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Before concluding this section, we connect the above discussion to the relationship between local-positivity and separability. As mentioned earlier, the cone of local positive operators and the cone of separable operators are dual to one another. The operator subspace we are dealing with is spanned by the orthonormal operators $\frac{1}{\sqrt{8}}\mathbb{I}$, $\frac{1}{\sqrt{6}}({{V}_{AB}}-\mathbb{I}/2)$, $\frac{1}{\sqrt{6}}({{V}_{AC}}-\mathbb{I}/2)$, and $\frac{1}{\sqrt{6}}({{V}_{BC}}-\mathbb{I}/2)$ with coordinates $\frac{1}{\sqrt{8}}$, $\sqrt{\frac{3}{8}}{{\eta }_{AB}}$, $\sqrt{\frac{3}{8}}{{\eta }_{AC}}$, and $\sqrt{\frac{3}{8}}{{\eta }_{BC}}$, respectively. In theorem 3.4, we determined the algebraic surface bounding the local positive operators; hence, the dual to this surface will bound the separable operators within this space. The dual to the Roman surface is known as Cayleyʼs cubic surface [51], which, for a given scale parameter w is characterized by

$$\begin{eqnarray*}\left| \begin{array}{ccccccccccccccc} w & x & y \\ x & w & z \\ y & z & w \\ \end{array} \right|=0.\end{eqnarray*}$$

We first set $x=\sqrt{\frac{3}{8}}{{\eta }_{AB}}$, $y=\sqrt{\frac{3}{8}}{{\eta }_{AC}}$, and $z=\sqrt{\frac{3}{8}}{{\eta }_{BC}}$. Then we must set w so that the Cayley surface delimits the separable states. For each extremal separable state in our space, there is a corresponding local-positive operator acting as an entanglement witness; a state is separable if the inner product with its entanglement witness is nonnegative.

Consider the extremal local-positive operator ${{\eta }_{AB}}={{\eta }_{AC}}={{\eta }_{BC}}=2/3$ that we made use of previously. This operator will act as an entanglement witness for another operator with ${{\eta }_{AB}}={{\eta }_{AC}}={{\eta }_{BC}}=\sigma$. We obtain σ by solving

$$\begin{eqnarray*}&&\left[ \begin{array}{ccccccccccccccc} \frac{1}{\sqrt{8}} \\ \sqrt{\frac{3}{8}}\frac{2}{3} \\ \sqrt{\frac{3}{8}}\frac{2}{3} \\ \sqrt{\frac{3}{8}}\frac{2}{3} \\ \end{array} \right]\cdot \left[ \begin{array}{ccccccccccccccc} \frac{1}{\sqrt{8}} \\ \sqrt{\frac{3}{8}}\sigma \\ \sqrt{\frac{3}{8}}\sigma \\ \sqrt{\frac{3}{8}}\sigma \\ \end{array} \right]=0,\end{eqnarray*}$$

to arrive at $\sigma =-\frac{1}{6}$. With this, the only value of w allowing the Cayley surface to be solved by $\sigma =-\frac{1}{6}$ is $w=\frac{1}{\sqrt{24}}$. Setting the scaling value and evaluating the determinant, we find that the separable states are bound by the surface

$$\begin{eqnarray*}&&\begin{array}{l} 1+54{{\eta }_{AB}}{{\eta }_{AC}}{{\eta }_{BC}}-9{{({{\eta }_{AB}}+{{\eta }_{AC}}+{{\eta }_{BC}})}^{2}} \\ \quad +18({{\eta }_{AB}}{{\eta }_{AC}}+{{\eta }_{AB}}{{\eta }_{BC}}+{{\eta }_{AC}}{{\eta }_{BC}})\geqslant 0. \\ \end{array}\end{eqnarray*}$$

This inequality may also be obtained using theorem 1 in [52]. The shape of the separable states is depicted in figure 7.

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Several remarks may be made. First, the set of separable states exhibits the tetrahedral symmetry shared by the classical joinability boundary and the local-positive joinability boundary. Thus, among the various boundaries we have considered in this three dimensional Euclidean space, the state- and channel-positive boundaries are the only ones not obeying tetrahedral symmetry. However, both the convex hull and the intersection of the state- and channel-positive cones bound regions which recover this symmetry. Indeed, it is a curious observation that the convex hull of these cones is 'nearly' the local-positive region, while the intersection is 'nearly' the set of separable states. Earlier we found, in the two-qudit case, that local-positivity coincides with the union of the state- and channel-positive regions, as well as that the separable region was their intersection. Here we consider the analog for three qubits. The result is that (i) the convex hull of state- and channel-positive operators is strictly contained in the set of local-positive operators; and (ii) the intersection of the state- and channel-positive operators is strictly contained in the set of separable states.

We may further interpret the latter result in terms of PPT considerations. The operators which result from a homocorrelation-mapped channel are necessarily PPT. Corollary 1 in [52] states that Werner operators which are positive and PPT are bi-separable. Thus, any state-positive operator which is also a homocorrelation mapped channel is necessarily bi-separable. Hence, the intersection of the four cones will be the set of states which are bi-separable with respect to any of the three partitions. This set clearly contains the set of tri-separable states. These observations illuminate the relationships among entanglement, quantum states, and quantum channels. Specifically, the homocorrelation map allows us to place quantum channels in the same arena as quantum states, and hence to directly compare and contrast them. Finding that the tri-separable operators are a proper subset of the bi-separable ones, we wonder what features these strictly bi-separable operators possess, and what does bi-separability imply for the states or channels supporting such correlations.