One-shot entanglement distillation beyond local operations and classical communication

We begin our work in section 2 with a brief introduction to a family of entanglement monotones ${T}_{{ \mathcal S }}^{(m)}$ which will play an important role in the later investigation of entanglement distillation. We characterise their properties and in particular show that the class of monotones generalises two known measures of entanglement—the robustness of entanglement and a distance-based quantifier based on trace distance—which will allow us to endow the measures with a direct operational meaning.

Our characterisation of entanglement distillation begins in section 3 where we establish explicit general connections between the quantifiers ${T}_{{ \mathcal S }}^{(m)}$, quantum hypothesis testing, and one-shot entanglement distillation through convex duality. The methods will form the foundations of the framework developed in this work.

We commence the explicit applications of our framework in section 3.1 by quantifying the distillation capabilities of several classes of operations based on the set of PPT states, recovering previous results of [25, 26] as well as describing new classes of operations in this context. The results additionally allow for an understanding of important asymptotic quantities, such as the regularised PPT relative entropy of entanglement or the Rains bound [24, 25], not just as bounds for distillable entanglement but as quantities with a precise operational meaning of their own. This section serves also as an introduction to the formalism considered in the manuscript and showcases the generality of our methods.

In section 3.2, we consider the class of separability-preserving operations [18, 27]. By relating the achievable fidelity of distillation with the monotones ${T}_{{ \mathcal S }}^{(m)}$ again, we establish an operational interpretation of the generalised robustness of entanglement in the context of distillation. Furthermore, we demonstrate a general operational equivalence in the distillation from pure states: all sets of operations, ranging from one-way LOCC to SEPP and PPT-preserving operations, achieve exactly the same performance in one-shot pure-state distillation. Although such an equivalence in the asymptotic regime was already known [10, 30], the correspondence already in the one-shot setting is remarkable, considering that the one-shot manipulation power of the larger sets of operations is generally much greater than that of LOCC. The results allow us to explicitly relate the fidelity of distillation of any pure state with an analytically computable norm of its Schmidt coefficients and express the computation of the $\varepsilon$-error one-shot distillable entanglement of a pure state as a convex quadratically-constrained linear program.

We continue in section 3.3 by establishing a similar operational equivalence in the distillation of isotropic states, showing that any class of operations ranging from separable operations to PPT- and separability-preserving operations achieve the same one-shot rates of distillation. Analogously, in section 3.4 we show that separability-preserving operations provide no advantage over PPT operations in the distillation from maximally correlated states, and furthermore, by relating the entanglement monotones with measures of quantum coherence, the achievable rates and fidelities of distillation can be computed efficiently as semidefinite programs.

In section 3.5 we show how our results immediately imply that in the setting of environment-assisted entanglement distillation [17, 31, 32], all considered operations—from one-way LOCC to PPT- and separability-preserving—achieve exactly the same performance. We furthermore recover the one-shot characterisation of [17] in a simplified fashion by employing the formalism introduced herein.

We conclude in section 3.6 with a discussion of zero-error distillation under the different sets of operations, obtaining in particular a single-letter formula for the asymptotic zero-error distillable entanglement under Rains-preserving operations which recovers a bound of [33] and endows it with an operational interpretation as a zero-error Rains bound.

An overview of the sets of operations considered in this work as well as their relative power in one-shot entanglement distillation is provided in figure 1 and table 1.

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Table 1.  Comparison between the one-shot $\varepsilon$-error entanglement distillation rates achievable under the different sets of operations considered in this work. We use > when the given inequality can be strict for some state, and $\geqslant$ if—to the best of our knowledge—the strictness of the inequality remains an open question.

General states	Pure states	Isotropic states	Max. corr. states
$\mathrm{SEPP}$	$\mathrm{SEPP}$	$\mathrm{SEPP}$	$\mathrm{SEPP}$
$\vee$	$\parallel$	$\parallel$	$\parallel$
${\mathrm{PPTP}}_{+}$	${\mathrm{PPTP}}_{+}$	${\mathrm{PPTP}}_{+}$	${\mathrm{PPTP}}_{+}$
$\vee$	$\parallel$	$\parallel$	$\parallel$
${\mathrm{PPTP}}_{+}^{\,^{\prime} }$	${\mathrm{PPTP}}_{+}^{\,^{\prime} }$	${\mathrm{PPTP}}_{+}^{\,^{\prime} }$	${\mathrm{PPTP}}_{+}^{\,^{\prime} }$
$\vee$	$\parallel$	$\parallel$	$\parallel$
$\mathrm{PPTP}$	$\mathrm{PPTP}$	$\mathrm{PPTP}$	$\mathrm{PPTP}$
$\parallel$	$\parallel$	$\parallel$	$\parallel$
$\mathrm{PPT}$	$\mathrm{PPT}$	$\mathrm{PPT}$	$\mathrm{PPT}$
$\vee$	$\parallel$	$\parallel$	$| \vee$
$\mathrm{SEP}$	$\mathrm{SEP}$	$\mathrm{SEP}$	$\mathrm{SEP}$
$| \vee$	$\parallel$	$| \vee$	$| \vee$
(1-)$\mathrm{LOCC}$	(1-)$\mathrm{LOCC}$	(1-)$\mathrm{LOCC}$	(1-)$\mathrm{LOCC}$

The analysis of this work will focus on understanding the achievable fidelity of distillation under different sets of operations, and establishing methods allowing us to relate it with convex optimisation problems which admit an efficient characterisation. To this end, we will introduce a family of entanglement monotones, which we will later explicitly endow with an operational interpretation and show to play a fundamental role in characterising entanglement distillation.

Consider a bipartite system shared between parties A and B, with d_A and d_B denoting the dimensions of the corresponding spaces. Let $d=\min \{{d}_{A},{d}_{B}\}$. We will consider the following sets of Hermitian matrices:

$$\begin{eqnarray}\begin{array}{r}\begin{array}{rcl}{\mathsf{P}}{\mathsf{P}}{\mathsf{T}} & = & \left\{\left.X\right|{\rm{T}}{\rm{r}}(X)=1,\,{X}^{{T}_{B}}\succcurlyeq 0\right\}\\ {{\mathsf{P}}{\mathsf{P}}{\mathsf{T}}}_{+} & = & \left\{\left.X\right|{\rm{T}}{\rm{r}}(X)=1,\,X\succcurlyeq 0,\,{X}^{{T}_{B}}\succcurlyeq 0\right\}\\ {\mathsf{S}}{\mathsf{E}}{\mathsf{P}} & = & {\rm{c}}{\rm{o}}{\rm{n}}{\rm{v}}\left\{\left.| \psi \rangle \langle \psi | \,\right|| \psi \rangle =| \phi { \rangle }_{A}\otimes | \eta { \rangle }_{B}\right\},\end{array}\end{array}\end{eqnarray} \tag{ 2 }$$

where ${X}^{{T}_{B}}$ is the partial transpose of X. Letting ${ \mathcal S }$ denote one of the above sets, a quantifier which found use in measuring the entanglement of quantum states in several contexts is the generalised robustness, defined as [34]

$$\begin{eqnarray}\begin{array}{r}\begin{array}{rcl}{R}_{{S}}^{{\mathbb{D}}}(\rho ) & = & min\left\{\left.\lambda \geqslant 0\right|\rho \preceq (\lambda +1)X,\,X\in {S}\right\}\\ & = & max\left\{\left. \langle \rho ,W \rangle \,\right|-{\mathbb{1}}\preceq W,\,W\in -{{S}}^{\ast }\right\}.\end{array}\end{array}\end{eqnarray} \tag{ 3 }$$

We then extend this definition to a class of measures

$$\begin{eqnarray}&&{T}_{{S}}^{(m)}(\rho ):= max\left\{\left. \langle \rho ,W \rangle \,\right|-{\mathbb{1}}\preceq W\preceq m{\mathbb{1}},\,W\in -{{S}}^{\ast }\right\}\end{eqnarray} \tag{ 4 }$$

for some parameter $m\in {{\mathbb{R}}}_{+}$. We note this class of measures has been considered in [35], but we have found that some of the results concerning the quantification and characterisation of ${T}_{{ \mathcal S }}^{(m)}$ stated there are in fact incorrect, so we present a self-contained investigation of their basic properties below and in the appendix.

A useful characterisation of the quantifiers is obtained by considering their dual form, which can be obtained as follows.

Proposition 1. The measures ${T}_{{ \mathcal S }}^{(m)}$ can be equivalently expressed as

$$\begin{eqnarray}&&{T}_{{ \mathcal S }}^{(m)}(\rho )=\min \left\{\left.m\mathrm{Tr}{\left(\rho -X\right)}_{+}+\mathrm{Tr}{\left(\rho -X\right)}_{-}\right|X\in { \mathcal S }* * \right\},\end{eqnarray} \tag{ 5 }$$

where ${\left(\rho -X\right)}_{+}$ (respectively, ${\left(\rho -X\right)}_{-}$) denotes the positive (negative) part of the Hermitian operator $\rho -X$.

The proof follows well-known methods in matrix analysis and we include it in the appendix for completeness.

In particular, for $m=1$ the measures take the remarkably simple form ${T}_{{ \mathcal S }}^{(1)}(\rho )={\min }_{X\in {{ \mathcal S }}^{* * }}{\parallel \rho -X\parallel }_{1}$. This quantity, considered first in the resource theory of coherence as the modified trace distance [36], generalises the commonly employed trace distance measure ${\min }_{\sigma \in { \mathcal S }}{\parallel \rho -\sigma \parallel }_{1}$. The reason why ${T}_{{ \mathcal S }}^{(1)}$ is a more suitable measure of entanglement than the trace distance itself is the fact that, contrary to ${T}_{{ \mathcal S }}^{(1)}$, the trace distance does not satisfy strong monotonicity under LOCC [36, 37] (i.e. the requirement that a measure M obeys $M(\rho )\geqslant {\sum }_{i}{p}_{i}M\left({{\rm{\Lambda }}}_{i}(\rho )\right)$ for any probabilistic protocol which applies an LOCC transformation ${{\rm{\Lambda }}}_{i}$ to ρ with probability p_i), which is often considered as one of the basic requirements that a measure of entanglement should satisfy [38]. This demonstrates a case where it becomes necessary to consider the distance with respect to the unnormalised cone ${ \mathcal S }* *$ rather than the set ${ \mathcal S }$ in order to ensure strong monotonicity.

Another interesting case is $m=d-1$, for which we obtain the following.

Proposition 2. For any ${ \mathcal S }\in \{{\mathsf{PPT}},{{\mathsf{PPT}}}_{+},{\mathsf{SEP}}\}$ it holds that

$$\begin{eqnarray}&&{T}_{{ \mathcal S }}^{(d-1)}(\rho )={R}_{{ \mathcal S }}^{{\mathbb{D}}}(\rho ).\end{eqnarray} \tag{ 6 }$$

Proof. Let $-W\in {\mathsf{SEP}}*$, and notice that ${\mathsf{SEP}}\subseteq {{\mathsf{PPT}}}_{+}\subseteq {\mathsf{PPT}}\Rightarrow {\mathsf{PPT}}* \subseteq {{\mathsf{PPT}}}_{+}^{* }\subseteq {\mathsf{SEP}}*$. Letting ${\lambda }_{\min }$ denote the smallest and ${\lambda }_{\max }$ the largest eigenvalue of a given matrix, we now use the property that $-W\in {\mathsf{SEP}}* \Rightarrow {\lambda }_{\min }(-W)\geqslant (1-d){\lambda }_{\max }(-W)$ [39, Cor. 5.5] together with the constraint $W\succcurlyeq -{\mathbb{1}}$ to obtain

$$\begin{eqnarray}&&{\lambda }_{\max }(W)\leqslant d-1.\end{eqnarray} \tag{ 7 }$$

It follows that the feasible sets for ${R}_{{ \mathcal S }}^{{\mathbb{D}}}$ and ${T}_{{ \mathcal S }}^{(d-1)}$ are equal and so the problems are equivalent. ■

From the above two Propositions, we have that the family of measures ${T}_{{ \mathcal S }}^{(m)}$ can be understood as interpolating between the robustness of ${ \mathcal S }$ for $m=d-1$, the modified trace distance for $m=1$, and the trivial value of 0 for $m=0$. It is furthermore easy to see that ${T}_{{ \mathcal S }}^{(m)}(\rho )={T}_{{ \mathcal S }}^{(d-1)}(\rho )$ for any $m\gt d-1$.

In the operational characterisation of entanglement distillation, it will be necessary to consider also generalisations of the above measures beyond sets of normalised (unit trace) Hermitian operators. To allow for this, we will now consider arbitrary compact sets of Hermitian operators ${ \mathcal Q }$ and define the quantity

$$\begin{eqnarray}&&{G}_{{ \mathcal Q }}^{(m)}(\rho ):= \sup \left\{\left\langle \rho ,W\right\rangle \left|0\preceq W\preceq {\mathbb{1}},\,W\in \displaystyle \frac{1}{m}{ \mathcal Q }^\circ \right.\right\}.\end{eqnarray} \tag{ 8 }$$

It is straightforward to see that if ${ \mathcal Q }$ is a set of unit trace operators, then ${G}_{{ \mathcal Q }}^{(m)}(\rho )$ is equal to $\tfrac{1}{m}\left({T}_{{ \mathcal Q }}^{(m-1)}(\rho )+1\right)$, although this relation does not hold in general.

To obtain a general dual formulation of ${G}_{{ \mathcal Q }}^{(m)}$, we will employ the formalism of gauge functions [40, 41]. The convex gauge function of a set ${ \mathcal Q }$ is defined as [40]

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{ \mathcal Q }}(\rho )=inf\left\{\left.\lambda \geqslant 0\right|\rho \in \lambda \,{\rm{c}}{\rm{o}}{\rm{n}}{\rm{v}}({ \mathcal Q })\right\}=sup\left\{\left. \langle \rho ,W \rangle \,\right|W\in {{ \mathcal Q }}^{\circ }\right\},\end{eqnarray} \tag{ 9 }$$

and can be thought of as an extension of the concept of a norm associated with a set—indeed, all norms are gauge functions, but the latter are more general. One can further notice that

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{{ \mathcal Q }}^{{}^{\circ }}}(\rho )=inf\left\{\left.\lambda \geqslant 0\right|\rho \in \lambda {{ \mathcal Q }}^{\circ }\right\}=sup\left\{\left. \langle \rho ,W \rangle \,\right|W\in {{ \mathcal Q }}^{\circ \circ }\right\},\end{eqnarray} \tag{ 10 }$$

where the first equality follows because the set ${ \mathcal Q }^\circ$ is always convex. We will take $\inf \varnothing =-\sup \varnothing =\infty$ and note that when the set ${ \mathcal Q }$ is compact, the infima and suprema in the definitions of the gauge functions are attained as long as they are finite. We then have the following.

Proposition 3. For any compact set ${ \mathcal Q }\subseteq {\mathbb{H}}$, we have that

$$\begin{eqnarray}&&{G}_{{ \mathcal Q }}^{(m)}(\rho )=\mathop{\inf }\limits_{Z\in {\mathbb{H}}}\,\mathrm{Tr}{\left(\rho -Z\right)}_{+}+\displaystyle \frac{1}{m}{{\rm{\Gamma }}}_{{ \mathcal Q }}(Z).\end{eqnarray} \tag{ 11 }$$

Proof. The definition of ${G}_{{ \mathcal Q }}^{(m)}$ imposes that $W\in ({-{\mathbb{H}}}_{+})^\circ \cap {\mathbb{D}}^\circ \cap (m{ \mathcal Q })^\circ =\left[({-{\mathbb{H}}}_{+})\cup {\mathbb{D}}\cup m\,\mathrm{conv}({ \mathcal Q })\right]^\circ$, from which we get ${G}_{{ \mathcal Q }}^{(m)}={{\rm{\Gamma }}}_{({-{\mathbb{H}}}_{+})\cup {\mathbb{D}}\cup m\mathrm{conv}({ \mathcal Q })}$. Since ${\mathbb{D}}$ and ${ \mathcal Q }$ are compact and ${{\mathbb{H}}}_{+}$ is closed, we get [40, Theorem 16.4]

$$\begin{eqnarray}\begin{array}{rcl}{G}_{{ \mathcal Q }}^{(m)}(\rho ) & = & \inf \left\{\left.{{\rm{\Gamma }}}_{{-{\mathbb{H}}}_{+}}(X)+{{\rm{\Gamma }}}_{{\mathbb{D}}}(Y)+{{\rm{\Gamma }}}_{m{ \mathcal Q }}(Z)\right|\rho =X+Y+Z\right\}\\ & = & \inf \left\{\left.{{\rm{\Gamma }}}_{{\mathbb{D}}}(Y)+{{\rm{\Gamma }}}_{m{ \mathcal Q }}(Z)\right|\rho =X+Y+Z,\,X\in {-{\mathbb{H}}}_{+}\right\}\\ & = & \inf \left\{\left.\mathrm{Tr}Y+\displaystyle \frac{1}{m}{{\rm{\Gamma }}}_{{ \mathcal Q }}(Z)\right|\rho -Z\preceq Y,\,Y\in {{\mathbb{H}}}_{+}\right\}=\mathrm{infTr}{\left(\rho -Z\right)}_{+}+\displaystyle \frac{1}{m}{{\rm{\Gamma }}}_{{ \mathcal Q }}(Z),\end{array}\end{eqnarray} \tag{ 12 }$$

where we have used that ${{\rm{\Gamma }}}_{{-{\mathbb{H}}}_{+}}(X)=0$ if $X\in {-{\mathbb{H}}}_{+}$ and $\infty$ otherwise, and ${{\rm{\Gamma }}}_{{\mathbb{D}}}(Y)=\mathrm{Tr}Y$ for any positive semidefinite Y and $\infty$ if Y is not positive semidefinite. ■

Remark. The above formula effectively constraints the optimisation to be over $Z\in { \mathcal Q }* *$, since for any $Z\notin { \mathcal Q }* *$, we have ${{\rm{\Gamma }}}_{{ \mathcal Q }}(Z)=\infty$. In particular, if ${ \mathcal Q }$ consists only of trace-one matrices, we can equivalently write

$$\begin{eqnarray}&&{G}_{{ \mathcal Q }}^{(m)}(\rho )=\mathop{\inf }\limits_{X\in { \mathcal Q }* * }\,\mathrm{Tr}{\left(\rho -X\right)}_{+}+\displaystyle \frac{1}{m}\mathrm{Tr}(X)\end{eqnarray} \tag{ 13 }$$

which reduces to the form in Proposition 1.

One of the first relaxations of LOCC in the literature was the class of separable operations (SEP) [23, 47], corresponding to all quantum channels ${\rm{\Lambda }}:{AB}\to A^{\prime} B^{\prime}$ whose Choi matrix is separable across the bipartition ${AA}^{\prime} | {BB}^{\prime}$. This set of maps has been shown to be strictly larger than LOCC [48], thus providing an upper bound on the capabilities of LOCC in distillation. However, the fact that the definition of $\mathrm{SEP}$ relies on the separability of the Choi matrix means that the set is not amenable to an efficient analytical characterisation, which then motivated the definitions of larger sets of operations. We begin with the investigation of several classes of such operations based on the set ${\mathsf{PPT}}$.

The class of PPT operations, due to Rains [24, 25], is defined to consist of all CPTP maps ${\rm{\Lambda }}:{AB}\to A^{\prime} B^{\prime}$ whose Choi matrix J_Λ satisfies ${J}_{{\rm{\Lambda }}}^{{T}_{{BB}^{\prime} }}\succcurlyeq 0$. In some works, a closely related class of 'PPT-preserving operations' has been considered [26, 49], motivated by the fact that $X\in {\mathsf{PPT}}\Rightarrow {\rm{\Lambda }}(X)\in {\mathsf{PPT}}$ for any PPT operation Λ. Although the two classes have sometimes been claimed to be equal, it is not difficult to see that only imposing the PPT-preserving constraint leads to a strictly larger class of quantum channels—consider, for instance, the channel which swaps subsystems A and B—so it is in fact incorrect to use the names 'PPT' and 'PPT-preserving' interchangeably when referring to operations. Interestingly, however, the two sets of channels lead to exactly the same rates of one-shot entanglement distillation (as well as dilution), as we will shortly see explicitly.

More recently, the name 'PPT-preserving operations' was also used to denote operations which map any PPT state to a PPT state, in the sense that $\sigma \in {{\mathsf{PPT}}}_{+}\Rightarrow {\rm{\Lambda }}(\sigma )\in {{\mathsf{PPT}}}_{+}$ [50]. It is well-known that this leads to a strictly larger class of operations than Rains' PPT operations [51], although an accurate way of referring to the class of PPT operations could be completely PPT-preserving [52], since the condition ${J}_{{\rm{\Lambda }}}^{{T}_{{BB}^{\prime} }}\succcurlyeq 0$ ensures the preservation of positivity when the map acts on a part of a larger system, akin to completely positive maps.

For clarity, let us begin with the precise definitions.

Definition 6. A CPTP map ${\rm{\Lambda }}:{AB}\to A^{\prime} B^{\prime}$ is called PPT if any one of the following equivalent conditions is satisfied [24].

• (i)  
  The Choi matrix J_Λ is PPT with respect to the partition ${AA}^{\prime} | {BB}^{\prime}$, i.e. ${J}_{{\rm{\Lambda }}}^{{T}_{{BB}^{\prime} }}\succcurlyeq 0$.
• (ii)  
  The map ${T}_{{B}^{{\prime} }}\,\circ \,{\rm{\Lambda }}\,\circ \,{T}_{B}$ is completely positive.
• (iii)  
  For any spaces $C,D$ such that ${d}_{C}={d}_{A}$ and ${d}_{D}={d}_{B}$ it holds that

  $$\begin{eqnarray*}&&\sigma \in {{\mathsf{PPT}}}_{+}({AC}| {BD})\Rightarrow {\rm{\Lambda }}\otimes \mathrm{id}(\sigma )\in {{\mathsf{PPT}}}_{+}({A}^{\prime} {C}| {B}^{\prime} {D})\end{eqnarray*}$$

  where $\mathrm{id}$ is the identity channel.

We will use $\mathrm{PPT}$ to denote the set of all such maps.

Definition 7. A CPTP map ${\rm{\Lambda }}:{AB}\to A^{\prime} B^{\prime}$ is called ${{\mathsf{PPT}}}_{+}$-preserving if

$$\begin{eqnarray}&&\sigma \in {{\mathsf{PPT}}}_{+}\Rightarrow {\rm{\Lambda }}(\sigma )\in {{\mathsf{PPT}}}_{+}.\end{eqnarray} \tag{ 23 }$$

We will use ${\mathrm{PPTP}}_{+}$ to denote the set of all such maps.

Definition 8. A CPTP map ${\rm{\Lambda }}:{AB}\to A^{\prime} B^{\prime}$ is called ${\mathsf{PPT}}$-preserving if either of the following equivalent conditions is satisfied:

• (i)  
  $X\in {\mathsf{PPT}}\Rightarrow {\rm{\Lambda }}(X)\in {\mathsf{PPT}}$.
• (ii)  
  The map ${T}_{{B}^{{\prime} }}\,\circ \,{\rm{\Lambda }}\,\circ \,{T}_{B}$ is positive, i.e. $X\in {{\mathbb{H}}}_{+}\Rightarrow {T}_{{B}^{{\prime} }}\,\circ \,{\rm{\Lambda }}\,\circ \,{T}_{B}(X)\in {{\mathbb{H}}}_{+}$.

We will use $\mathrm{PPTP}$ to denote the set of all such maps.

We now characterise the operational capabilities of the different sets of operations. Note that the fidelity of distillation under the class $\mathrm{PPT}$ has previously been obtained by Rains [25], and an explicit expression for the rate of distillation in terms of ${D}_{H}^{\varepsilon }$ appeared more recently in [20]. Reference [26] considered the class $\mathrm{PPTP}$ in this context, but the capabilities of ${\mathrm{PPTP}}_{+}$ have not been explicitly investigated before.

Theorem 9. The fidelity of distillation under the classes of operations $\mathrm{PPT}$, $\mathrm{PPTP}$, and ${\mathrm{PPTP}}_{+}$ is given by

$$\begin{eqnarray}&&\begin{array}{l}{F}_{{\mathrm{PPTP}}_{+}}(\rho ,m)={G}_{{{\mathsf{PPT}}}_{+}}^{(m)}(\rho )\\ {F}_{\mathrm{PPT}}(\rho ,m)={F}_{\mathrm{PPTP}}(\rho ,m)={G}_{{\mathsf{PPT}}^{\prime} }^{(m)}(\rho ),\end{array}\end{eqnarray} \tag{ 24 }$$

where ${\mathsf{PPT}}^{\prime} =\mathrm{conv}({\mathsf{PPT}}\cup -{\mathsf{PPT}})=\left\{\left.X\right|{\parallel {X}^{{T}_{B}}\parallel }_{1}\leqslant 1\right\}$, and hence the one-shot distillable entanglement can be expressed as

$$\begin{eqnarray}&&{E}_{d,{\mathrm{PPTP}}_{+}}^{(1),\varepsilon }(\rho )={\left\lfloor \mathop{\min }\limits_{\sigma \in {{\mathsf{PPT}}}_{+}}{D}_{H}^{\varepsilon }(\rho \parallel \sigma )\right\rfloor }_{\mathrm{log}}\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&{E}_{d,\mathrm{PPT}}^{(1),\varepsilon }(\rho )={E}_{d,\mathrm{PPTP}}^{(1),\varepsilon }(\rho )={\left\lfloor \mathop{\min }\limits_{X\in {\mathsf{PPT}}^{\prime} }{D}_{H}^{\varepsilon }(\rho \parallel X)\right\rfloor }_{\mathrm{log}}.\end{eqnarray} \tag{ 26 }$$

Proof. Since ${{\rm{\Psi }}}_{m}$ is invariant under any unitary of the form $U\otimes {U}^{* }$, it is in particular invariant under the twirling ${ \mathcal T }(\cdot ):= \int (U\otimes {U}^{* })\cdot {\left(U\otimes {U}^{* }\right)}^{\dagger }\,{\rm{d}}U$, where the integration is performed with respect to the Haar measure of the unitary group. We can then without loss of generality consider only trace-preserving operations of the form ${\rm{\Lambda }}={ \mathcal T }\,\circ \,{\rm{\Lambda }}$, giving

$$\begin{eqnarray}&&{\rm{\Lambda }}(Z)=\left\langle Z,X\right\rangle {{\rm{\Psi }}}_{m}+\displaystyle \frac{\left\langle Z,{\mathbb{1}}-X\right\rangle }{{m}^{2}-1}\left({\mathbb{1}}-{{\rm{\Psi }}}_{m}\right)\end{eqnarray} \tag{ 27 }$$

as this is the most general form of an operator invariant under twirling [53]. Since ${{\rm{\Psi }}}_{m}^{{T}_{B}}=\tfrac{1}{m}({P}_{m}^{+}-{P}_{m}^{-})$, where ${P}_{m}^{+}$ (respectively, ${P}_{m}^{-}$) denote the projector onto the symmetric (antisymmetric) subspace, we have

$$\begin{eqnarray}&&{\rm{\Lambda }}{\left(Z\right)}^{{T}_{B}}=\displaystyle \frac{{P}_{m}^{+}}{m+1}\left(\displaystyle \frac{\mathrm{Tr}Z}{m}+\left\langle Z,X\right\rangle \right)+\displaystyle \frac{{P}_{m}^{-}}{m-1}\left(\displaystyle \frac{\mathrm{Tr}Z}{m}-\left\langle Z,X\right\rangle \right).\end{eqnarray} \tag{ 28 }$$

Using the mutual orthogonality of ${P}_{m}^{\pm }$, we obtain the general conditions

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Lambda }}(Z)\succcurlyeq 0 & \ \Longleftrightarrow \ & \left\langle Z,X\right\rangle \geqslant 0\\ {\rm{\Lambda }}{\left(Z\right)}^{{T}_{B}}\succcurlyeq 0 & \ \Longleftrightarrow \ & -\displaystyle \frac{\mathrm{Tr}Z}{m}\leqslant \left\langle Z,X\right\rangle \leqslant \displaystyle \frac{\mathrm{Tr}Z}{m}.\end{array}\end{eqnarray} \tag{ 29 }$$

Noting in addition that the complete positivity of Λ imposes $0\preceq X\preceq {\mathbb{1}}$, we can constrain the map Λ such that ${\rm{\Lambda }}(\sigma )\in {{\mathsf{PPT}}}_{+}$ for any $\sigma \in {{\mathsf{PPT}}}_{+}$ to get

$$\begin{eqnarray}&&{F}_{{\mathrm{PPTP}}_{+}}(\rho ,m)=\max \left\{\left\langle \rho ,X\right\rangle \left|0\preceq X\preceq {\mathbb{1}},\,\left\langle \sigma ,X\right\rangle \leqslant \displaystyle \frac{1}{m}\,\forall \sigma \in {{\mathsf{PPT}}}_{+}\right.\right\},\end{eqnarray} \tag{ 30 }$$

and similarly, by imposing that $S\in {\mathsf{PPT}}\Rightarrow {\rm{\Lambda }}(S)\in {\mathsf{PPT}}$ we have

$$\begin{eqnarray}\begin{array}{rcl}{F}_{\mathrm{PPTP}}(\rho ,m) & = & \max \left\{\left\langle \rho ,X\right\rangle \left|0\preceq X\preceq {\mathbb{1}},\,\left|\left\langle S,X\right\rangle \right|\leqslant \displaystyle \frac{1}{m}\,\forall S\in {\mathsf{PPT}}\right.\right\}\\ & = & \max \left\{\left\langle \rho ,X\right\rangle \left|0\preceq X\preceq {\mathbb{1}},\,\left\langle S,X\right\rangle \leqslant \displaystyle \frac{1}{m}\,\forall S\in {\mathsf{PPT}}^{\prime} \right.\right\}.\end{array}\end{eqnarray} \tag{ 31 }$$

Noting that the Choi matrix of the map (27) is given by ${J}_{{\rm{\Lambda }}}={X}_{{AB}}\otimes {{\rm{\Psi }}}_{{m}_{{A}^{{\prime} }{B}^{{\prime} }}}+\tfrac{1}{{m}^{2}-1}{\left({\mathbb{1}}-X\right)}_{{AB}}\otimes {\left({\mathbb{1}}-{{\rm{\Psi }}}_{m}\right)}_{{A}^{{\prime} }{B}^{{\prime} }}$, an explicit computation yields

$$\begin{eqnarray}&&{J}_{{\rm{\Lambda }}}^{{T}_{{BB}^{\prime} }}\succcurlyeq 0\ \Longleftrightarrow \ -\displaystyle \frac{1}{m}{\mathbb{1}}\preceq {X}^{{T}_{B}}\preceq \displaystyle \frac{1}{m}{\mathbb{1}},\end{eqnarray} \tag{ 32 }$$

which is precisely the condition $\left\langle S,X\right\rangle \leqslant \tfrac{1}{m}\,\forall S\in {\mathsf{PPT}}^{\prime}$, yielding the equality between ${F}_{d,\mathrm{PPTP}}(\rho ,m)$ and ${F}_{d,\mathrm{PPT}}(\rho ,m)$.

Since the distillation fidelities ${F}_{{\mathrm{PPTP}}_{+}}$ and ${F}_{\mathrm{PPTP}}$ are precisely of the form ${G}_{{{\mathsf{PPT}}}_{+}}$ and ${G}_{{\mathsf{PPT}}^{\prime} }$, respectively, the result follows by Theorem 5.■

The Theorem establishes an operational equivalence between the sets of operations $\mathrm{PPT}$ and $\mathrm{PPTP}$, although we stress again that in fact $\mathrm{PPT}\subsetneq \mathrm{PPTP}$: in particular, the swap operation, defined as ${\rm{\Lambda }}(| {ij}\rangle \langle {kl}| )=| {ji}\rangle \langle {lk}|$ in a basis and extended by linearity, trivially preserves the positivity of the partial transpose of any operator, while the partial transpose of the Choi matrix J_Λ can be verified to be non-positive. This can be understood by noting that the swap operation does not preserve PPT states when acting only on a part of a larger system—indeed, if Alice and Bob each possess a singlet and exchange only half of it, they will have generated (maximal) entanglement. Notice also that we have explicitly shown a difference between the distillation rates of $\mathrm{PPTP}$ and ${\mathrm{PPTP}}_{+}$, thus immediately implying that $\mathrm{PPTP}\subsetneq {\mathrm{PPTP}}_{+}$.

In addition, we recall an argument in [50] which investigated a gap between $\mathrm{PPT}$ and ${\mathrm{PPTP}}_{+}$ operations by showing that the negativity (a known monotone under $\mathrm{PPT}$ [54]) can increase under ${\mathrm{PPTP}}_{+}$. This argument no longer applies to $\mathrm{PPTP}$—the negativity can be expressed as a robustness-type quantifier with respect to the set ${\mathsf{PPT}}$ [54] and it follows straightforwardly that this is a strong monotone under $\mathrm{PPTP}$ [41]. The gap between $\mathrm{PPTP}$ and $\mathrm{PPT}$ is therefore much more subtle.

Although it is not easy to characterise the asymptotic rates of distillation under $\mathrm{PPT}$ and $\mathrm{PPTP}$ maps, we have the following characterisation of distillable entanglement under ${\mathrm{PPTP}}_{+}$, thus establishing a limit on the asymptotic performance of $\mathrm{PPT}$ and $\mathrm{PPTP}$ (see also [55]).

Corollary 10. The asymptotic distillable entanglement under ${\mathrm{PPTP}}_{+}$ is given by the regularised relative entropy of entanglement with respect to the set ${{\mathsf{PPT}}}_{+}$,

$$\begin{eqnarray}&&{E}_{d,{\mathrm{PPTP}}_{+}}^{\infty }(\rho )={E}_{R,{{\mathsf{PPT}}}_{+}}^{\infty }(\rho ):= \mathop{\mathrm{lim}}\limits_{n\to \infty }\mathop{\min }\limits_{\sigma \in {{\mathsf{PPT}}}_{+}}\displaystyle \frac{1}{n}D({\rho }^{\otimes n}\parallel \sigma )\end{eqnarray} \tag{ 33 }$$

with D denoting the quantum relative entropy.

Proof. Follows directly from the generalised quantum Stein's Lemma [56], which shows precisely that the regularisation of the hypothesis testing relative entropy ${\min }_{\sigma \in {{\mathsf{PPT}}}_{+}}{D}_{H}^{\varepsilon }(\rho \parallel \sigma )$ in the asymptotic limit with $\varepsilon$ going to 0 is given by the regularised relative entropy. ■

We remark that, although ${E}_{d,{\mathrm{PPTP}}_{+}}^{\infty }(\rho )$ is not known in general, it has been computed exactly for classes of all orthogonally invariant states (including isotropic and Werner states) [57], and it has been shown that there exist states such that ${E}_{d,\mathrm{PPT}}^{\infty }(\rho )\lt {E}_{d,{\mathrm{PPTP}}_{+}}^{\infty }(\rho )$ [58].

3.1.1. Rains set and distillation

The above Corollary in particular gives an operational interpretation to the regularised relative entropy ${E}_{R,{{\mathsf{PPT}}}_{+}}^{\infty }$, introduced first as a bound for distillable entanglement in [24]. One can then wonder whether similar operational interpretation can be given to other asymptotic quantities in entanglement distillation theory. We will show that it is indeed the case for one of the most fundamental of such bounds, the regularised Rains bound [25, 33, 57], constituting the tightest known bound for the asymptotically distillable entanglement. It is defined as

$$\begin{eqnarray}&&{E}_{{\rm{Rains}}}^{\infty }(\rho ):= \mathop{\mathrm{lim}}\limits_{n\to \infty }\mathop{\min }\limits_{X\in {{\mathsf{PPT}}}_{+}^{{\prime} }}\displaystyle \frac{1}{n}D({\rho }^{\otimes n}\parallel X),\end{eqnarray} \tag{ 34 }$$

where ${{\mathsf{PPT}}}_{+}^{{\prime} }=\left\{\left.X\succcurlyeq 0\right|{\parallel {X}^{{T}_{B}}\parallel }_{1}\leqslant 1\right\}$ is the so-called Rains set. To relate this quantity with the distillation of entanglement, we will define the class of Rains-preserving operations ${\mathrm{PPTP}}_{+}^{\,^{\prime} }$ as all maps such that $X\in {{\mathsf{PPT}}}_{+}^{{\prime} }\Rightarrow {\rm{\Lambda }}(X)\in {{\mathsf{PPT}}}_{+}^{{\prime} }$. We then have the following.

Theorem 11. The fidelity of distillation and one-shot distillable entanglement under Rains-preserving operations ${\mathrm{PPTP}}_{+}^{\,^{\prime} }$ are given by

$$\begin{eqnarray}\begin{array}{rcl}{F}_{{\mathrm{PPTP}}_{+}^{\prime }}(\rho ,m) & = & {G}_{{{\mathsf{PPT}}}_{+}^{{\prime} }}^{(m)}(\rho )\\ {E}_{d,{\mathrm{PPTP}}_{+}^{\,^{\prime} }}^{(1),\varepsilon }(\rho ) & = & {\left\lfloor \mathop{\min }\limits_{X\in {{\mathsf{PPT}}}_{+}^{{\prime} }}{D}_{H}^{\varepsilon }(\rho \parallel X)\right\rfloor }_{\mathrm{log}}.\end{array}\end{eqnarray} \tag{ 35 }$$

Proof. The proof proceeds analogously to Theorem 9. The crucial step is to notice that for the isotropic operator ${\rm{\Lambda }}(Z)=\left\langle Z,X\right\rangle {{\rm{\Psi }}}_{m}+\tfrac{\left\langle Z,{\mathbb{1}}-X\right\rangle }{{m}^{2}-1}\left({\mathbb{1}}-{{\rm{\Psi }}}_{m}\right)$ we have

$$\begin{eqnarray}&&{\parallel {\rm{\Lambda }}{\left(Z\right)}^{{T}_{B}}\parallel }_{1}=\displaystyle \frac{m}{2}\left|\displaystyle \frac{\mathrm{Tr}Z}{m}+\left\langle Z,X\right\rangle \right|+\displaystyle \frac{m}{2}\left|\displaystyle \frac{\mathrm{Tr}Z}{m}-\left\langle Z,X\right\rangle \right|,\end{eqnarray} \tag{ 36 }$$

where we have used that $\tfrac{1}{m+1}{\parallel {P}_{m}^{+}\parallel }_{1}=\tfrac{1}{m-1}{\parallel {P}_{m}^{-}\parallel }_{1}=\tfrac{m}{2}$ and that ${P}_{m}^{\pm }$ are mutually orthogonal projections. For any $Z\in {{\mathsf{PPT}}}_{+}^{{\prime} }$, it is then easy to verify that we have ${\parallel {\rm{\Lambda }}{\left(Z\right)}^{{T}_{B}}\parallel }_{1}\leqslant 1\ \Longleftrightarrow \ \left\langle Z,X\right\rangle \leqslant \tfrac{1}{m}$. This gives

$$\begin{eqnarray}&&{F}_{{\mathrm{PPTP}}_{+}^{\prime }}(\rho ,m)=\max \left\{\left\langle \rho ,X\right\rangle \left|0\preceq X\preceq {\mathbb{1}},\,\left\langle Z,X\right\rangle \leqslant \displaystyle \frac{1}{m}\,\forall Z\in {{\mathsf{PPT}}}_{+}^{{\prime} }\right.\right\}={G}_{{{\mathsf{PPT}}}_{+}^{{\prime} }}^{(m)}(\rho ).\end{eqnarray} \tag{ 37 }$$

The statement about ${E}_{d,{\mathrm{PPTP}}_{+}^{\,^{\prime} }}^{(1),\varepsilon }$ then follows directly from Theorem 5. ■

Once again, an application of the generalised quantum Stein's Lemma [56] then gives

$$\begin{eqnarray}&&{E}_{d,{\mathrm{PPTP}}_{+}^{\,^{\prime} }}^{\infty }(\rho )={E}_{{\rm{Rains}}}^{\infty }(\rho ),\end{eqnarray} \tag{ 38 }$$

which establishes an explicit operational interpretation of the regularised Rains bound as the asymptotic rate of entanglement distillation under the class of Rains-preserving operations. Noting that ${\mathrm{PPTP}}_{+}^{\,^{\prime} }\supseteq \mathrm{PPTP}$ by definition, we recover the result that ${E}_{{\rm{Rains}}}^{\infty }$ upper bounds the asymptotic distillable entanglement under $\mathrm{PPT}$ operations [25]. It is interesting to conjecture that we have equality between ${E}_{d,\mathrm{PPT}}^{\infty }$ and ${E}_{d,{\mathrm{PPTP}}_{+}^{\,^{\prime} }}^{\infty }$ (see [20]), but we were not able to establish this.

To obtain a tighter bound on distillable entanglement, one could then ask about distillation under operations which completely preserve the Rains set, in the sense that

$$\begin{eqnarray}&&X\in {{\mathsf{PPT}}}_{+}^{{\prime} }({AC}| {BD})\Rightarrow {\rm{\Lambda }}\otimes \mathrm{id}(X)\in {{\mathsf{PPT}}}_{+}^{{\prime} }(A^{\prime} C| B^{\prime} D)\end{eqnarray} \tag{ 39 }$$

for some spaces $C,D$ such that ${d}_{C}={d}_{A}$ and ${d}_{D}={d}_{B}$. We will call any such channel completely Rains-preserving. In other words, a map ${\rm{\Lambda }}:{AB}\to A^{\prime} B^{\prime}$ is completely Rains-preserving iff it is CPTP and

$$\begin{eqnarray}&&{\parallel {\rm{\Lambda }}\otimes \mathrm{id}{\left(X\right)}^{{T}_{{B}^{{\prime} }{D}^{{\prime} }}}\parallel }_{1}\leqslant 1\quad \forall \,X:X\succcurlyeq 0,{\parallel {X}^{{T}_{{BD}}}\parallel }_{1}\leqslant 1.\end{eqnarray} \tag{ 40 }$$

We will now show that these maps are precisely the set of PPT channels.

Theorem 12. A quantum channel is PPT iff it is completely Rains-preserving.

Proof. One direction is straightforward: if Λ is completely positive and completely Rains-preserving, then for any $\sigma \in {{\mathsf{PPT}}}_{+}({AC}| {BD})$ we necessarily have ${\rm{\Lambda }}\otimes \mathrm{id}(\sigma )\in {{\mathsf{PPT}}}_{+}(A^{\prime} C^{\prime} | B^{\prime} D^{\prime} )$ due to the fact that ${{\mathsf{PPT}}}_{+}={{\mathsf{PPT}}}_{+}^{{\prime} }\cap {{\mathbb{H}}}_{+}$. This means that Λ is completely ${{\mathsf{PPT}}}_{+}$-preserving, i.e. PPT.

To see the opposite inclusion, define a 'PPT-diamond norm' of any map Γ as

$$\begin{eqnarray*}\begin{array}{rcl}{\parallel {\rm{\Gamma }}\parallel }_{\unicode{x029EB}} & := & \max \left\{\left.{\parallel {\rm{\Gamma }}\otimes \mathrm{id}({X}^{{T}_{{BD}}})\parallel }_{1}\right|{\parallel {X}^{{T}_{{BD}}}\parallel }_{1}\leqslant 1,\,X\succcurlyeq 0\right\}\\ & = & \max \left\{\left.{\parallel {\rm{\Gamma }}\otimes \mathrm{id}(X)\parallel }_{1}\right|{\parallel X\parallel }_{1}\leqslant 1,\,{X}^{{T}_{{BD}}}\succcurlyeq 0\right\}.\end{array}\end{eqnarray*}$$

Rewriting equation (40) one can see that Λ is completely Rains-preserving iff

$$\begin{eqnarray*}&&{\parallel {T}_{{B}^{{\prime} }}\circ {\rm{\Lambda }}\circ {T}_{B}\parallel }_{\unicode{x029EB}}\leqslant 1.\end{eqnarray*}$$

Notice then that, for any Hermiticity-preserving map Γ it holds that

$$\begin{eqnarray*}&&{\parallel {\rm{\Gamma }}\parallel }_{\unicode{x029EB}}\leqslant \max \left\{\left.{\parallel {\rm{\Gamma }}\otimes \mathrm{id}(X)\parallel }_{1}\right|{\parallel X\parallel }_{1}\leqslant 1\right\}={\parallel {\rm{\Gamma }}\parallel }_{\diamond },\end{eqnarray*}$$

where ${\parallel \cdot \parallel }_{\diamond }$ is the diamond norm (completely bounded trace norm) [59, 60]. Since for any PPT channel Λ the map ${T}_{{B}^{{\prime} }}\circ \,{\rm{\Lambda }}\,\circ \,{T}_{B}$ is CPTP, we have that any PPT channel satisfies ${\parallel {T}_{{B}^{{\prime} }}\circ {\rm{\Lambda }}\circ {T}_{B}\parallel }_{\diamond }=1$ [60] and therefore is completely Rains-preserving.■

The above result establishes an operational connection between the sets ${\mathrm{PPTP}}_{+}$ and ${\mathrm{PPTP}}_{+}^{\,^{\prime} }$, showing that their 'completely preserving' variants reduce to the same set of operations (PPT).

The class of separability-preserving operations $\mathrm{SEPP}$ is defined as all CPTP maps Λ such that $\sigma \in {\mathsf{SEP}}\Rightarrow {\rm{\Lambda }}(\sigma )\in {\mathsf{SEP}}$, that is, as the maximal class of free (non-entangling) operations in the resource theory of entanglement. Notice that this class does not completely preserve separability, in the sense that it could generate entanglement if applied to a part of a larger system; if such complete preservation is imposed, we instead recover the class of separable operations. The inclusions between the different classes of operations are shown in figure 1.

The fidelity of distillation under $\mathrm{SEPP}$ was first derived in [27], and can be used to characterise the distillable entanglement as follows.

Lemma 13 It holds that ${F}_{\mathrm{SEPP}}(\rho ,m)={G}_{{\mathsf{SEP}}}^{(m)}(\rho )=\tfrac{1}{m}\left({T}_{{\mathsf{SEP}}}^{(m-1)}(\rho )+1\right)$, and hence

$$\begin{eqnarray}&&{E}_{d,\mathrm{SEPP}}^{(1),\varepsilon }(\rho )={\left\lfloor \mathop{\min }\limits_{\sigma \in {\mathsf{SEP}}}{D}_{H}^{\varepsilon }(\rho \parallel \sigma )\right\rfloor }_{\mathrm{log}}.\end{eqnarray} \tag{ 41 }$$

Proof. Follows in exactly the same way as the proof of Theorem 9, since isotropic states of the form

$$\begin{eqnarray}&&\left\langle Z,X\right\rangle {{\rm{\Psi }}}_{m}+\displaystyle \frac{\left\langle Z,{\mathbb{1}}-X\right\rangle }{{m}^{2}-1}\left({\mathbb{1}}-{{\rm{\Psi }}}_{m}\right)\end{eqnarray} \tag{ 42 }$$

are separable if and only if they are PPT [53]. ■

By the inclusion ${\mathsf{SEP}}\subseteq {{\mathsf{PPT}}}_{+}\subset {\mathsf{PPT}}^{\prime}$, we immediately have that

$$\begin{eqnarray}&&{F}_{\mathrm{SEPP}}(\rho ,m)\geqslant {F}_{{\mathrm{PPTP}}_{+}}(\rho ,m)\geqslant {F}_{\mathrm{PPT}}(\rho ,m)\end{eqnarray} \tag{ 43 }$$

thus establishing a hierarchy of rates of distillation between the operations $\mathrm{SEPP}$, ${\mathrm{PPTP}}_{+}$, and $\mathrm{PPT}$. Notice that this does not follow from their definition, as there is no inclusion between the sets of maps $\mathrm{SEPP}$ and $\mathrm{PPT}$, nor between $\mathrm{SEPP}$ and ${\mathrm{PPTP}}_{+}$.

Crucially, for any pure state, the fidelity of distillation can be computed exactly. To establish this result, we will employ the so-called m-distillation norm, introduced in [61] as

$$\begin{eqnarray}&&{\parallel | x \rangle \parallel }_{[m]}:=\mathop{min}\limits_{| x \rangle =| y \rangle +| z \rangle }{\parallel | y \rangle \parallel }_{{\ell }_{1}}+\sqrt{m}{\parallel | z \rangle \parallel }_{{\ell }_{2}}=max\left\{\left.\left| \langle x| w \rangle \right|\,\right|{\parallel | w \rangle \parallel }_{{\ell }_{{\rm{\infty }}}}\leqslant 1,\,{\parallel | w \rangle \parallel }_{{\ell }_{2}}\leqslant \sqrt{m}\right\}\end{eqnarray} \tag{ 44 }$$

for any vector $| x\rangle \in {{\mathbb{C}}}^{d}$. One can immediately notice from the inequality ${\parallel \cdot \parallel }_{{{\ell }}_{2}}\leqslant {\parallel \cdot \parallel }_{{{\ell }}_{1}}\leqslant \sqrt{d}{\parallel \cdot \parallel }_{{{\ell }}_{2}}$ that we have ${\parallel | x\rangle \parallel }_{[1]}={\parallel | x\rangle \parallel }_{{{\ell }}_{2}}$ and ${\parallel | x\rangle \parallel }_{[d]}={\parallel | x\rangle \parallel }_{{{\ell }}_{1}}$. Notably, for any normalised vector $| x\rangle \in {{\mathbb{C}}}^{d}$ and any integer $m\in \{1,\ldots ,d\}$, the norm admits an exact expression as [61]

$$\begin{eqnarray*}&&{\parallel | x\rangle \parallel }_{[m]}={\parallel | {x}_{1:m-{k}^{\star }}^{\downarrow }\rangle \parallel }_{{{\ell }}_{1}}+\sqrt{{k}^{\star }}{\parallel | {x}_{m-{k}^{\star }+1:d}^{\downarrow }\rangle \parallel }_{{{\ell }}_{2}},\end{eqnarray*}$$

where $| {x}_{1:k}^{\downarrow }\rangle$ denotes the vector consisting of the k largest (by magnitude) coefficients of $| x\rangle$, analogously $| {x}_{k+1:d}^{\downarrow }\rangle$ denotes the vector of the d − k smallest coefficients of $| x\rangle$ with $| {x}_{1:0}^{\downarrow }\rangle$ being the zero vector, and we define

$$\begin{eqnarray}&&{k}^{\star }:= \mathop{\arg \,\min }\limits_{1\leqslant k\leqslant m}\displaystyle \frac{1}{k}{\parallel | {x}_{m-k+1:d}^{\downarrow }\rangle \parallel }_{{{\ell }}_{2}}^{2}.\end{eqnarray} \tag{ 45 }$$

We stress that the computation of ${\parallel | x\rangle \parallel }_{[m]}$ is thus reduced to evaluating $m-1$ inequalities.

We will now use $| {\xi }_{\psi }\rangle \in {{\mathbb{R}}}^{d}$ to denote the vector of Schmidt coefficients of a pure state $| \psi \rangle$, in the sense that $| {\xi }_{\psi }\rangle ={\left({\alpha }_{1},\ldots ,{\alpha }_{d}\right)}^{T}$ where $| \psi \rangle ={\sum }_{i}{\alpha }_{i}| i{\rangle }_{A}| i{\rangle }_{B}$ for some orthonormal bases $\{| {i}_{A}\rangle \},\{| i{\rangle }_{B}\}$. Employing the m-distillation norm, we then have the following.

Theorem 14. For any $m\geqslant 1$, it holds that

$$\begin{eqnarray}&&{T}_{{\mathsf{SEP}}}^{(m-1)}(\psi )={\parallel | {\xi }_{\psi }\rangle \parallel }_{\,[m]}^{2}-1,\end{eqnarray} \tag{ 46 }$$

and in particular ${F}_{\mathrm{SEPP}}(\psi ,m)=\tfrac{1}{m}{\parallel | {\xi }_{\psi }\rangle \parallel }_{\,[m]}^{2}$.

Proof. To begin, notice that with a simple rearrangement of terms ${T}_{{\mathsf{SEP}}}^{(m-1)}$ can be written as

$$\begin{eqnarray}&&max\left\{\left. \langle \rho ,W \rangle \,\right|0\preceq W\preceq m{\mathbb{1}},\,W\in {{\mathsf{S}}{\mathsf{E}}{\mathsf{P}}}^{\circ }\right\}=max\left\{ \langle \rho ,W \rangle \left|W\succcurlyeq 0,\,W\in {\left({\mathsf{S}}{\mathsf{E}}{\mathsf{P}}\cup \displaystyle \frac{1}{m}{\mathbb{D}}\right)}^{{}^{\circ }}\right.\right\},\end{eqnarray} \tag{ 47 }$$

where we used that $({ \mathcal C }\cup { \mathcal D })^\circ ={ \mathcal C }^\circ \cap { \mathcal D }^\circ$ for convex and closed sets. The set $\mathrm{conv}({\mathsf{SEP}}\cup \tfrac{1}{m}{\mathbb{D}})=: {{ \mathcal Q }}_{m}$ can be noticed to be the convex hull of rank-one terms as ${{ \mathcal Q }}_{m}=\mathrm{conv}\left\{\left.| x\rangle \langle x| \right|| x\rangle \in { \mathcal V }\cup {{ \mathcal N }}_{m}\right\}$ where ${ \mathcal V }=\{| \phi {\rangle }_{A}\otimes | \eta {\rangle }_{B}\}$ is the set of all normalised product state vectors and ${{ \mathcal N }}_{m}:= \left\{\left.| x\rangle \right|{\parallel | x\rangle \parallel }_{{{\ell }}_{2}}=1/\sqrt{m}\right\}.$

By Theorem 10 in [41], for any pure state $| \psi \rangle$ we then have⁶

$$\begin{eqnarray}\begin{array}{r}\begin{array}{rcl}{T}_{{\mathsf{S}}{\mathsf{E}}{\mathsf{P}}}^{(m-1)}(\psi )+1 & = & max\left\{\left. \langle \psi | W| \psi \rangle \,\right|W\succcurlyeq 0,W\in {{ \mathcal Q }}_{m}^{{}^{\circ }}\right\}\\ & = & max\left\{\left.{\left| \langle \psi | w \rangle \right|}^{2}\,\right|\,| w \rangle \in ({V}\cup {{N}}_{m}{)}{}^\circ \right\}={{\rm{\Gamma }}}_{{V}\cup {{N}}_{m}}{\left(| \psi \rangle \right)}^{2}\end{array}\end{array}\end{eqnarray} \tag{ 48 }$$

which means that the value of ${T}_{{\mathsf{SEP}}}^{(m-1)}$ will be given by a corresponding gauge function ${{\rm{\Gamma }}}_{{ \mathcal V }\cup {{ \mathcal N }}_{m}}{\left(| \psi \rangle \right)}^{2}$ defined at the level of the underlying Hilbert space, instead of the whole space of Hermitian operators. Since ${ \mathcal V }$ and ${{ \mathcal N }}_{m}$ are both compact sets, by standard results in convex analysis (see e.g. [40], 16.4.1 and 15.1.2), this gauge can be obtained as

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{ \mathcal V }\cup {{ \mathcal N }}_{m}}(| \psi \rangle )=\mathop{\min }\limits_{| \psi \rangle =| x\rangle +| y\rangle }{{\rm{\Gamma }}}_{{ \mathcal V }}(| x\rangle )+{{\rm{\Gamma }}}_{{{ \mathcal N }}_{m}}(| y\rangle ).\end{eqnarray} \tag{ 49 }$$

Now, for any vector $| x\rangle$ we have ${{\rm{\Gamma }}}_{{{ \mathcal N }}_{m}}(| x\rangle )=\sqrt{m}{\parallel x\parallel }_{{{\ell }}_{2}}=\sqrt{m}{\parallel | {\xi }_{x}\rangle \parallel }_{{{\ell }}_{2}}$ and it is known that ${{\rm{\Gamma }}}_{{ \mathcal V }}(| x\rangle )$ can be computed as ${\parallel | {\xi }_{x}\rangle \parallel }_{{{\ell }}_{1}}$ (see e.g. [62, 63]). By optimising over vectors $| x\rangle ,| y\rangle$ in the Schmidt basis of $| \psi \rangle$ only, the problem reduces to the m-distillation norm of the Schmidt vector $| {\xi }_{\psi }\rangle$, and we thus have ${T}_{{\mathsf{SEP}}}^{(m-1)}(\psi )\leqslant {\parallel | {\xi }_{\psi }\rangle \parallel }_{\,[m]}^{2}-1$.

To show the opposite inequality, we use the fact that $({ \mathcal V }\cup {{ \mathcal N }}_{m})^\circ ={ \mathcal V }^\circ \cap {{ \mathcal N }}_{m}^{^\circ }$ to write the gauge ${{\rm{\Gamma }}}_{{ \mathcal V }\cup {{ \mathcal N }}_{m}}$ in its dual form as

$$\begin{eqnarray}&&{T}_{{\mathsf{S}}{\mathsf{E}}{\mathsf{P}}}^{(m-1)}(\psi )+1=max\left\{\left.{\left| \langle \psi | x \rangle \right|}^{2}\,\right|\,{{\rm{\Gamma }}}_{{{N}}_{m}}^{{}^{\circ }}(| x \rangle )\leqslant 1,\,{{\rm{\Gamma }}}_{{V}}^{{}^{\circ }}(| x \rangle )\leqslant 1\right\},\end{eqnarray} \tag{ 50 }$$

where we have, for any $| x\rangle$, ${{\rm{\Gamma }}}_{{ \mathcal V }}^{^\circ }(| x\rangle )={\parallel | {\xi }_{x}\rangle \parallel }_{{{\ell }}_{\infty }}$ [64]. By optimising over all vectors $| x\rangle$ in the Schmidt basis of $| \psi \rangle$, we recover again the m-distillation norm of $| {\xi }_{\psi }\rangle$ and the result follows.■

From the above, we then have the fidelity of distillation under $\mathrm{SEPP}$ of a pure state $| \psi \rangle$ as $\tfrac{1}{m}{\parallel | {\xi }_{\psi }\rangle \parallel }_{\,[m]}^{2}$. Crucially, the m-distillation norm can be closely connected with the concept of majorisation, allowing us to relate it to the optimal fidelity of pure-state distillation under LOCC and one-way LOCC ($1\mbox{--}\mathrm{LOCC}$), which was previously considered in [65]. We will now rederive the exact expression for the fidelity of pure-state distillation under LOCC in terms of the m-distillation norm, and in particular establish an operational equivalence between all relevant sets of operations in the distillation of entanglement from pure states.

Theorem 15. For any pure state $| \psi \rangle$, any integer $m\geqslant 1$, and any set of operations ${ \mathcal O }\in \{1\mbox{--}\mathrm{LOCC},\mathrm{LOCC},\mathrm{PPT},{\mathrm{PPTP}}_{+}^{\,^{\prime} },{\mathrm{PPTP}}_{+},\mathrm{SEPP}\}$, the fidelity of distillation is given by

$$\begin{eqnarray}&&{F}_{{ \mathcal O }}(\psi ,m)=\displaystyle \frac{1}{m}{\parallel | {\xi }_{\psi }\rangle \parallel }_{\,[m]}^{2}.\end{eqnarray} \tag{ 51 }$$

Proof. We begin by recalling that the m-distillation norm of $| {\xi }_{\psi }\rangle := {\left({\alpha }_{1},\ldots ,{\alpha }_{d}\right)}^{T}$ can be computed as

$$\begin{eqnarray}&&{\parallel | {\xi }_{\psi }\rangle \parallel }_{[m]}={\parallel | {\alpha }_{1:m-{k}^{\star }}^{\downarrow }\rangle \parallel }_{{{\ell }}_{1}}+\sqrt{{k}^{\star }}{\parallel | {\alpha }_{m-{k}^{\star }+1:d}^{\downarrow }\rangle \parallel }_{{{\ell }}_{2}},\end{eqnarray} \tag{ 52 }$$

with ${k}^{\star }:= \mathop{\arg \,\min }\limits_{1\leqslant k\leqslant m}\tfrac{{\parallel | {\alpha }_{m-k+1:d}^{\downarrow }\rangle \parallel }_{{{\ell }}_{2}}^{2}}{k}$.

Now, it is well-known that the deterministic transformation from $| \psi \rangle$ to another pure state $| \eta \rangle$ is possible with (either one-way and two-way) $\mathrm{LOCC}$ if and only if the Schmidt vector $| {\xi }_{\psi }\rangle ={\left({\alpha }_{1},\ldots ,{\alpha }_{d}\right)}^{T}$ is majorised by the Schmidt vector $| {\xi }_{\eta }\rangle := {\left({\beta }_{1},\ldots ,{\beta }_{d}\right)}^{T}$ [66], that is

$$\begin{eqnarray}&&\displaystyle \sum _{i=1}^{k}{\alpha }_{i}^{2}\leqslant \displaystyle \sum _{i=1}^{k}{\beta }_{i}^{2}\quad \forall k\in \{1,\ldots ,d\},\end{eqnarray} \tag{ 53 }$$

where we have assumed without loss of generality that the Schmidt coefficients are given in non-increasing order. Let us then define the ansatz

$$\begin{eqnarray}&&| \eta \rangle =\displaystyle \sum _{i=1}^{m-{k}^{\star }}{\alpha }_{i}| {ii}\rangle +\displaystyle \sum _{i=m-{k}^{\star }+1}^{m}\displaystyle \frac{{\parallel | {\alpha }_{m-{k}^{\star }+1:d}^{\downarrow }\rangle \parallel }_{{{\ell }}_{2}}}{\sqrt{{k}^{\star }}}| {ii}\rangle \end{eqnarray} \tag{ 54 }$$

expressed in the Schmidt basis of $| {{\rm{\Psi }}}_{m}\rangle$, where ${k}^{\star }$ is defined as above. To see that the Schmidt coefficients of $| \eta \rangle$ majorise the ones of $| \psi \rangle$, let us assume that ${k}^{\star }\gt 1$ (as otherwise the desired relation is trivial) and consider the following chain of equivalent inequalities:

$$\begin{eqnarray}&&\begin{array}{r}\begin{array}{l}\displaystyle \frac{{\parallel | {\alpha }_{m-{k}^{\star }+1:d}^{\downarrow } \rangle \parallel }_{{\ell }_{2}}^{2}}{{k}^{\star }}\leqslant \displaystyle \frac{{\parallel | {\alpha }_{m-{k}^{\star }+2:d}^{\downarrow } \rangle \parallel }_{{\ell }_{2}}^{2}}{{k}^{\star }-1}\\ {\alpha }_{m-{k}^{\star }+1}^{2}+{\parallel | {\alpha }_{m-{k}^{\star }+2:d}^{\downarrow } \rangle \parallel }_{{\ell }_{2}}^{2}\leqslant \displaystyle \frac{{k}^{\star }}{{k}^{\star }-1}{\parallel | {\alpha }_{m-{k}^{\star }+2:d}^{\downarrow } \rangle \parallel }_{{\ell }_{2}}^{2}\\ {\alpha }_{m-{k}^{\star }+1}^{2}\leqslant \displaystyle \frac{1}{{k}^{\star }-1}\left(1-{\parallel | {\alpha }_{1:m-{k}^{\star }+1}^{\downarrow } \rangle \parallel }_{{\ell }_{2}}^{2}\right)\\ {k}^{\star }{\alpha }_{m-{k}^{\star }+1}^{2}\leqslant \left(1-{\parallel | {\alpha }_{1:m-{k}^{\star }}^{\downarrow } \rangle \parallel }_{{\ell }_{2}}^{2}\right)\\ {\alpha }_{m-{k}^{\star }+1}^{2}\leqslant \displaystyle \frac{{\parallel | {\alpha }_{m-{k}^{\star }+1:d}^{\downarrow } \rangle \parallel }_{{\ell }_{2}}^{2}}{{k}^{\star }},\end{array}\end{array}\end{eqnarray} \tag{ 55 }$$

where the first line follows by definition of ${k}^{\star }$, and in the third and fifth lines we have used the fact that $| \psi \rangle$ is a normalised pure state. Thus, we have

$$\begin{eqnarray}&&{F}_{1\mbox{--}\mathrm{LOCC}}(\psi ,m)\geqslant {\left|\left\langle {{\rm{\Psi }}}_{m}| \eta \right\rangle \right|}^{2}=\displaystyle \frac{1}{m}{\left({\parallel {\alpha }_{1:m-{k}^{\star }}^{\downarrow }\parallel }_{{{\ell }}_{1}}+\sqrt{{k}^{\star }}{\parallel {\alpha }_{m-{k}^{\star }+1:d}^{\downarrow }\parallel }_{{{\ell }}_{2}}\right)}^{2}=\displaystyle \frac{1}{m}{\parallel | {\xi }_{\psi }\rangle \parallel }_{\,[m]}^{2}.\end{eqnarray} \tag{ 56 }$$

On the other hand, since $1\mbox{--}\mathrm{LOCC}\subseteq \mathrm{SEPP}$, we have

$$\begin{eqnarray}&&{F}_{1\mbox{--}\mathrm{LOCC}}(\psi ,m)\leqslant {F}_{\mathrm{SEPP}}(\psi ,m)=\displaystyle \frac{1}{m}\left({T}_{{\mathsf{SEP}}}^{(m-1)}(\psi )+1\right)=\displaystyle \frac{1}{m}{\parallel | {\xi }_{\psi }\rangle \parallel }_{\,[m]}^{2}\end{eqnarray} \tag{ 57 }$$

by Theorem 14, which concludes the proof.■

The exact correspondence is rather surprising. The operations PPT and SEPP are known to be much more powerful than LOCC in general—indeed, $\mathrm{SEPP}$ exhibit no bound entanglement whatsoever [27], and even in the manipulation of pure states PPT operations can, for instance, arbitrarily increase the Schmidt rank (number of Schmidt coefficients) of a pure state [52, 67], which cannot increase whatsoever under $\mathrm{LOCC}$ or $\mathrm{SEP}$ [12]. The result then shows that even such large sets cannot outperform one-way LOCC in entanglement distillation from pure states, even in the one-shot setting.

Using the above expression, we can furthermore show that the computation of ${E}_{d,{ \mathcal O }}^{(1),\varepsilon }$ belongs to a class of efficiently solvable optimisation problems known as quadratically-constrained linear programs [68].

Corollary 16. For any set of operations ${ \mathcal O }\in \{1\mbox{--}\mathrm{LOCC},\mathrm{LOCC},\mathrm{PPT},{\mathrm{PPTP}}_{+}^{\,^{\prime} },{\mathrm{PPTP}}_{+},\mathrm{SEPP}\}$, the one-shot distillable entanglement of a pure state can be expressed exactly as the optimal value of the convex quadratically-constrained linear program

$$\begin{eqnarray}&&{E}_{d,{ \mathcal O }}^{(1),\varepsilon }(\psi )={\left\lfloor -\mathrm{log}\min \left\{{\parallel | \omega \rangle \parallel }_{{{\ell }}_{\infty }}^{2}| \left\langle {\xi }_{\psi }| \omega \right\rangle \geqslant \sqrt{1-\varepsilon },{\parallel | \omega \rangle \parallel }_{{{\ell }}_{2}}\leqslant 1,| \omega \rangle \in {{\mathbb{R}}}_{+}^{d}\right\}\right\rfloor }_{\mathrm{log}}.\end{eqnarray} \tag{ 58 }$$

Proof. The dual form of the m-distillation norm, which we recall here as

$$\begin{eqnarray}&&{\parallel | x \rangle \parallel }_{[m]}=max\left\{\left.\left.\left| \langle x| w \rangle \,\right.\right|\right|{\parallel | w \rangle \parallel }_{{\ell }_{{\rm{\infty }}}}\leqslant 1,\,{\parallel | w \rangle \parallel }_{{\ell }_{2}}\leqslant \sqrt{m}\right\},\end{eqnarray} \tag{ 59 }$$

gives

$$\begin{eqnarray}\begin{array}{rcl}{E}_{d,{ \mathcal O }}^{(1),\varepsilon }(\psi ) & = & \mathrm{log}\,\max \left\{m\in {\mathbb{N}}\left|\displaystyle \frac{1}{m}{\parallel | {\xi }_{\psi }\rangle \parallel }_{\,[m]}^{2}\geqslant 1-\varepsilon \right.\right\}\\ & = & {\left\lfloor -\mathrm{log}\min \left\{{\parallel | \omega \rangle \parallel }_{{{\ell }}_{\infty }}^{2}| {\left|\left\langle {\xi }_{\psi }| \omega \right\rangle \right|}^{2}\geqslant 1-\varepsilon ,{\parallel | \omega \rangle \parallel }_{{{\ell }}_{2}}\leqslant 1\right\}\right\rfloor }_{\mathrm{log}}\end{array}\end{eqnarray} \tag{ 60 }$$

and we conclude by noting that it suffices to optimise over vectors with non-negative coefficients since $| {\xi }_{\psi }\rangle$ is also non-negative. ■

The above result can be compared with the bounds obtained for LOCC distillable entanglement in [16, 17], and in fact we have tightened the bounds to an exact expression for the one-shot distillable entanglement:

$$\begin{eqnarray}&&{E}_{d,{ \mathcal O }}^{(1),\varepsilon }(\psi )={\left\lfloor -\mathrm{log}\min \left\{\left.{\parallel {\rho }_{B}\parallel }_{\infty }\right|F\left({\rho }_{B},{\mathrm{Tr}}_{A}(\psi )\right)\geqslant 1-\varepsilon \right\}\right\rfloor }_{\mathrm{log}}.\end{eqnarray} \tag{ 61 }$$

The Theorem also leads to an interesting characterisation of the m-distillation norm in two different ways. Notice that the proof of Theorem 14 in fact shows that the m-distillation norm of the Schmidt vector $| {\xi }_{x}\rangle$ of a vector $| x\rangle$ can be equivalently written as a norm at the level of the vector $| x\rangle$ itself:

$$\begin{eqnarray}\begin{array}{rcl}{\parallel | {\xi }_{x}\rangle \parallel }_{[m]} & = & \mathop{\min }\limits_{| x\rangle =| y\rangle +| z\rangle }{\parallel | {\xi }_{y}\rangle \parallel }_{{{\ell }}_{1}}+\sqrt{m}{\parallel | {\xi }_{z}\rangle \parallel }_{{{\ell }}_{2}}\\ & = & \max \left\{\left.\left|\left\langle x| w\right\rangle \right|\right|{\parallel | {\xi }_{w}\rangle \parallel }_{{{\ell }}_{\infty }}\leqslant 1,\,{\parallel | {\xi }_{w}\rangle \parallel }_{{{\ell }}_{2}}\leqslant \sqrt{m}\right\}.\end{array}\end{eqnarray} \tag{ 62 }$$

Writing $| \psi \rangle \mathop{\longrightarrow }\limits^{\mathrm{LOCC}}| \eta \rangle$ to denote that the pure state transformation is possible with $\mathrm{LOCC}$, we then have

$$\begin{eqnarray}\begin{array}{r}\begin{array}{rcl}{\parallel | {\xi }_{\psi } \rangle \parallel }_{[m]} & = & \sqrt{m}\,max\left\{\left| \langle {{\rm{\Psi }}}_{m}| \eta \rangle \right|\left|\,| \psi \rangle \mathop{\longrightarrow }\limits^{{\rm{L}}{\rm{O}}{\rm{C}}{\rm{C}}}| \eta \rangle \right.\right\}\\ & = & \sqrt{m}\,max\left\{\left| \langle \psi | \eta \rangle \right|\left|\,| \eta \rangle \mathop{\longrightarrow }\limits^{{\rm{L}}{\rm{O}}{\rm{C}}{\rm{C}}}| {{\rm{\Psi }}}_{m} \rangle \right.\right\},\end{array}\end{array}\end{eqnarray} \tag{ 63 }$$

where the maximisation is over normalised state vectors $| \eta \rangle$, and the second line is precisely equation (62).

As a straightforward Corollary of the results above, we can establish the value of the quantifiers ${G}_{{ \mathcal S }}^{(m)}$ for several sets other than ${\mathsf{SEP}}$.

Corollary 17. For any $1\leqslant m\leqslant d$, any pure state $| \psi \rangle$, and any ${ \mathcal S }\in \{{\mathsf{SEP}},{\mathsf{PPT}},{{\mathsf{PPT}}}_{+},{\mathsf{PPT}}^{\prime} \}$ we have

$$\begin{eqnarray}&&{G}_{{ \mathcal S }}^{(m)}(| \psi \rangle \langle \psi | )=\displaystyle \frac{1}{m}{\parallel | {\xi }_{\psi }\rangle \parallel }_{\,[m]}^{2}.\end{eqnarray} \tag{ 64 }$$

Going beyond pure states, combined with Proposition 2 the expression for ${F}_{\mathrm{SEPP}}(\rho ,d)$ gives a direct operational interpretation to the generalised robustness of entanglement ${R}_{{\mathsf{SEP}}}^{{\mathbb{D}}}$ by showing that

$$\begin{eqnarray}&&{R}_{{\mathsf{SEP}}}^{{\mathbb{D}}}(\rho )={{dF}}_{\mathrm{SEPP}}(\rho ,d)-1\end{eqnarray} \tag{ 65 }$$

for any state ρ, and in fact by Theorem 9 also for the class ${\mathrm{PPTP}}_{+}$ we have the relation ${R}_{{{\mathsf{PPT}}}_{+}}^{{\mathbb{D}}}(\rho )={{dF}}_{{\mathrm{PPTP}}_{+}}(\rho ,d)-1$. This complements the known operational applications of this quantity [18, 69, 70]. Note also that all of the other measures in the family ${T}_{{\mathsf{SEP}}}^{(m)}$ and ${T}_{{{\mathsf{PPT}}}_{+}}^{(m)}$, including the modified trace distance quantifiers, are given similar interpretations.

An important use of the fidelity of distillation ${F}_{\mathrm{LOCC}}(\rho ,m)$ in the particular case $m=d$ is as the fidelity of teleportation, that is, the best average fidelity one can achieve in the task of quantum teleportation by employing an LOCC protocol on the state ρ [71]. Notably, in [72] it was then shown that for ${d}_{A}={d}_{B}=2$, we have ${F}_{\mathrm{LOCC}}(\rho ,d)={F}_{\mathrm{PPT}}(\rho ,d)$ showing that even PPT protocols (or SEPP protocols, by Lemma 13) cannot enhance the fidelity of teleportation of the given state. By Theorem 15, we know that this relation extends to all pure states in all dimensions; that is, ${F}_{\mathrm{LOCC}}(\psi ,d)={F}_{\mathrm{PPT}}(\psi ,d)={F}_{\mathrm{SEPP}}(\psi ,d)$.

Remark. In [35] (Proposition 9) it was claimed that the asymptotic distillable entanglement ${E}_{d,\mathrm{LOCC}}^{\infty }(\rho )$ of any state is upper bounded by $\mathrm{log}\left({T}_{{\mathsf{SEP}}}^{(m)}(\rho )+1\right)$ for any $m\geqslant 1$. This is clearly not true, as ${T}_{{\mathsf{SEP}}}^{(m)}(\rho )\leqslant m\forall \rho$ and the distillable entanglement obeys no such restriction. We have seen, however, that the quantifiers ${T}_{{\mathsf{SEP}}}^{(m)}(\rho )$ characterise exactly the fidelity of distillation.

As a side note, noticing the similarity between the distillation under ${{\mathsf{PPT}}}_{+}$-preserving, Rains-preserving, and ${\mathsf{SEP}}$-preserving operations, it might appear that the hypothesis testing relative entropy ${D}_{H}^{\varepsilon }$ in general quantifies the rate of distillation under a set of operations which is defined to preserve a given set of operators. This claim is supported by recent independent results concerned with distillation in a class of general quantum resources [73], but it does not hold in full generality as the distillation under $\mathrm{PPTP}$ operations shows (see Theorem 9), and indeed also distillation in the resource theory of coherence [61] is a counterexample.

Consider $d={d}_{A}={d}_{B}$ and define the isotropic states as

$$\begin{eqnarray}&&{\rho }_{f}=f{{\rm{\Psi }}}_{d}+\displaystyle \frac{1-f}{{d}^{2}-1}\left({\mathbb{1}}-{{\rm{\Psi }}}_{d}\right)\end{eqnarray} \tag{ 66 }$$

with $0\leqslant f\leqslant 1$. This class of states is particularly useful due to its strong symmetry, allowing for a much easier evaluation of their entanglement properties [53]. We then have the following result, showing the operational equivalence of all sets of channels from $\mathrm{SEP}$ to ${\mathrm{PPTP}}_{+}$ and $\mathrm{SEPP}$ in distiling entanglement from isotropic states, and extending the known characterisation of isotropic state distillation under PPT operations considered in [25].

Theorem 18. For any isotropic state ${\rho }_{f}$ and any ${ \mathcal O }\in \{\mathrm{SEP},\mathrm{PPT},{\mathrm{PPTP}}_{+}^{\,^{\prime} },{\mathrm{PPTP}}_{+},\mathrm{SEPP}\}$, it holds that

$$\begin{eqnarray}{F}_{{ \mathcal O }}({\rho }_{f},m)=\left\{\begin{array}{lc}\displaystyle \frac{1}{m} & f\leqslant \displaystyle \frac{1}{d}\\ \displaystyle \frac{{df}-1}{d-1}+\displaystyle \frac{d(1-f)}{m(d-1)} & f\geqslant \displaystyle \frac{1}{d}.\end{array}\right.\end{eqnarray} \tag{ 67 }$$

Proof. Take $1\leqslant m\leqslant d$. If $f\leqslant \tfrac{1}{d}$, then ${\rho }_{f}\in {\mathsf{SEP}}$, and we have ${F}_{\mathrm{SEP}}({\rho }_{f},m)={F}_{\mathrm{PPT}}({\rho }_{f},m)={F}_{\mathrm{SEPP}}({\rho }_{f},m)\,=\tfrac{1}{m};$ we will therefore assume that $f\geqslant \tfrac{1}{d}$ in the sequel.

Recall by our previous arguments that, due to twirling, we can limit ourselves to considering operations of the form

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{W}(Z)=\left\langle Z,W\right\rangle {{\rm{\Psi }}}_{m}+\displaystyle \frac{\left\langle Z,{\mathbb{1}}-W\right\rangle }{{m}^{2}-1}\left({\mathbb{1}}-{{\rm{\Psi }}}_{m}\right),\end{eqnarray} \tag{ 68 }$$

and the fidelity of distillation under a set ${ \mathcal O }$ is then given by

$$\begin{eqnarray}&&{F}_{{ \mathcal O }}({\rho }_{f},m)=\max \left.\left\{\left\langle {\rho }_{f},W\right\rangle \right|{{\rm{\Lambda }}}_{W}\in { \mathcal O }\right\}.\end{eqnarray} \tag{ 69 }$$

Here, noting the invariance of ${\rho }_{f}$ under twirling, we can twirl once more; in particular, $\left\langle {\rho }_{f},W\right\rangle =\left\langle { \mathcal T }({\rho }_{f}),{ \mathcal T }(W)\right\rangle =\left\langle {\rho }_{f},{ \mathcal T }(W)\right\rangle \,\forall W$, so we can again limit the considered W to be of the form

$$\begin{eqnarray}&&W=\alpha {{\rm{\Psi }}}_{d}+\beta {\mathbb{1}},\end{eqnarray} \tag{ 70 }$$

where $0\leqslant \beta \leqslant 1$, $0\leqslant \alpha +\beta \leqslant 1$. The Choi operator of the corresponding map ${{\rm{\Lambda }}}_{W}$ is then of the form

$$\begin{eqnarray}&&{J}_{{{\rm{\Lambda }}}_{W}}=\displaystyle \frac{\alpha {m}^{2}}{{m}^{2}-1}{{\rm{\Psi }}}_{{dm}}+\displaystyle \frac{\beta {m}^{2}-1}{{m}^{2}-1}{{\mathbb{1}}}_{{AB}}\otimes {{\rm{\Psi }}}_{m}-\displaystyle \frac{\alpha }{{m}^{2}-1}{{\rm{\Psi }}}_{d}\otimes {{\mathbb{1}}}_{{A}^{{\prime} }{B}^{{\prime} }}+\displaystyle \frac{1-\beta }{{m}^{2}-1}{{\mathbb{1}}}_{{{ABA}}^{{\prime} }{B}^{{\prime} }},\end{eqnarray} \tag{ 71 }$$

where ${d}_{{A}^{{\prime} }}={d}_{{B}^{{\prime} }}=m$. By [74], Theorem 6, ${J}_{{{\rm{\Lambda }}}_{W}}\in {\mathsf{SEP}}({AA}^{\prime} :{BB}^{\prime} )$ if and only if the following conditions are all satisfied:

$$\begin{eqnarray}&&d-m\alpha +{dm}\beta \geqslant 0,\ d+m\alpha -{dm}\beta \geqslant 0,\ d-m\alpha -{dm}\beta \geqslant 0,\ m\alpha +{d}^{2}\,m\beta -d(\alpha +\beta )\geqslant 0.\end{eqnarray} \tag{ 72 }$$

Let us choose

$$\begin{eqnarray}&&W=\displaystyle \frac{d(m-1)}{(d-1)m}{{\rm{\Psi }}}_{d}+\displaystyle \frac{d-m}{(d-1)m}{\mathbb{1}}\end{eqnarray} \tag{ 73 }$$

for which the inequalities (72) can be readily verified to hold. This gives

$$\begin{eqnarray}&&{F}_{\mathrm{SEP}}({\rho }_{f},m)\geqslant \left\langle {\rho }_{f},W\right\rangle =\displaystyle \frac{{df}-1}{d-1}+\displaystyle \frac{d(1-f)}{m(d-1)}.\end{eqnarray} \tag{ 74 }$$

On the other hand, take

$$\begin{eqnarray}&&X=\displaystyle \frac{d(1-f)}{{d}^{2}-1}\left({{\rm{\Psi }}}_{d}+\displaystyle \frac{1}{d}{\mathbb{1}}\right).\end{eqnarray} \tag{ 75 }$$

It is known that ${{\rm{\Psi }}}_{d}+\tfrac{1}{d}{\mathbb{1}}\in {\mathsf{SEP}}* *$ [34], which gives

$$\begin{eqnarray}&&{F}_{\mathrm{SEPP}}({\rho }_{f},m)\leqslant \mathrm{Tr}{\left({\rho }_{f}-X\right)}_{+}+\displaystyle \frac{1}{m}\mathrm{Tr}X=\displaystyle \frac{{df}-1}{d-1}+\displaystyle \frac{d(1-f)}{m(d-1)},\end{eqnarray} \tag{ 76 }$$

where we used Lemma 13 together with the dual form of ${G}_{{\mathsf{SEP}}}^{(m)}$ from Proposition 3.■

We remark that the above also gives a general way of lower-bounding the fidelity of distillation under separable operations of any state with a simple linear program, tight for all isotropic states:

$$\begin{eqnarray}&&{F}_{\mathrm{SEP}}(\rho ,m)\geqslant \max \left\{\left.\alpha \left\langle \rho ,{{\rm{\Psi }}}_{d}\right\rangle +\beta \right|0\leqslant \beta \leqslant 1,\,0\leqslant \alpha +\beta \leqslant 1,\,\mathrm{equation}(72)\right\}.\end{eqnarray} \tag{ 77 }$$

Unsurprisingly, however, a numerical investigation reveals this bound to be rather ineffective beyond the set of isotropic states.

Note also that a general investigation of one- and multi-shot entanglement distillation from isotropic states under PPT operations as a linear program has been explored in [20, 25].

Let us consider a bipartite system with $d={d}_{A}={d}_{B}$. A maximally correlated state is any state of the form ${\rho }_{\mathrm{mc}}={\sum }_{i,j}{\rho }_{{ij}}| {ii}\rangle \langle {jj}|$ for some local orthonormal bases $\{| i\rangle \}$ [24]. The name for this class of states comes from the fact is that the two parties are guaranteed to obtain the same measurement results for any measurement in their local basis $\{| i\rangle \}$.

Notice that any maximally correlated state has a corresponding single-party state $\widetilde{{\rho }_{\mathrm{mc}}}:= {\sum }_{i,j}{\rho }_{{ij}}| i\rangle \langle j|$ with the same coefficients in an orthonormal basis $\{| i\rangle \}$. This led to comparisons between the manipulation of maximally correlated states and the resource theory of coherence, which studies the properties of superposition as a quantum resource [75, 76]. In particular, it has been conjectured in several works that the resource theory of coherence is equivalent to the resource theory of entanglement restricted to maximally correlated states [77, 78]. Although this conjecture is still unsolved in full generality, we know that many operational quantifiers such as the entanglement of formation, relative entropy of entanglement (and other Rényi entropy–based entanglement measures), and asymptotic distillable entanglement can be evaluated on maximally correlated states by quantifying the corresponding coherence quantifiers, typically significantly simpler to evaluate and satisfying useful properties such as additivity [77, 79]. Furthermore, an operational equivalence between transformations acting on $\widetilde{{\rho }_{\mathrm{mc}}}$ and LOCC operations acting on ${\rho }_{\mathrm{mc}}$ has been suggested, although so far this conjecture has been shown only in specific cases [77, 78].

To obtain a result allowing us to quantify the one-shot distillable entanglement of maximally correlated states, we will fix a choice of basis $\{| i\rangle \}{}_{i=1}^{d}$ for the single-party state $\widetilde{{\rho }_{\mathrm{mc}}}$, and use ${ \mathcal I }:= \,\mathrm{conv}\{| i\rangle \langle i| \}{}_{i}$ to denote the set of all incoherent (diagonal) states in this basis. Furthermore, we define the subset of separable states ${{\mathsf{SEP}}}_{\mathrm{mc}}:= \,\mathrm{conv}\{| {ii}\rangle \langle {ii}| \}{}_{i}$ where $\{| {ii}\rangle \}{}_{i=1}^{d}$ is the maximally correlated basis of the state ${\rho }_{\mathrm{mc}}$. We then get the following.

Theorem 19. For any maximally correlated state, any $m\geqslant 1$, and any choice of operations ${ \mathcal O }\in \{\mathrm{PPT},{\mathrm{PPTP}}_{+}^{\,^{\prime} },{\mathrm{PPTP}}_{+},\mathrm{SEPP}\}$ it holds that

$$\begin{eqnarray}&&{F}_{{ \mathcal O }}({\rho }_{\mathrm{mc}},m)={G}_{{ \mathcal I }}^{(m)}(\widetilde{{\rho }_{\mathrm{mc}}}).\end{eqnarray} \tag{ 78 }$$

Proof. Using Lemma 13 together with Proposition 3, we have

$$\begin{eqnarray}\begin{array}{rcl}{F}_{\mathrm{PPT}}({\rho }_{\mathrm{mc}},m) & \leqslant & {F}_{\mathrm{SEPP}}({\rho }_{\mathrm{mc}},m)=\mathop{\min }\limits_{X\in {\mathsf{SEP}}* * }\mathrm{Tr}{\left({\rho }_{\mathrm{mc}}-X\right)}_{+}+\displaystyle \frac{1}{m}\mathrm{Tr}\left(X\right)\\ & \leqslant & \mathop{\min }\limits_{X\in {\mathrm{SEP}}_{\mathrm{mc}}^{* * }}\mathrm{Tr}{\left({\rho }_{\mathrm{mc}}-X\right)}_{+}+\displaystyle \frac{1}{m}\mathrm{Tr}\left(X\right)\\ & = & \mathop{\min }\limits_{X\in { \mathcal I }* * }\mathrm{Tr}{\left(\widetilde{{\rho }_{\mathrm{mc}}}-X\right)}_{+}+\displaystyle \frac{1}{m}\mathrm{Tr}\left(X\right)={G}_{{ \mathcal I }}^{(m)}\left(\widetilde{{\rho }_{\mathrm{mc}}}\right).\end{array}\end{eqnarray} \tag{ 79 }$$

On the other hand, let ${W}^{\star }={\sum }_{i,j}{W}_{{ij}}| i\rangle \langle j|$ be the optimal solution to the dual problem of

$$\begin{eqnarray}&&{G}_{{I}}^{(m)}(\widetilde{{\rho }_{{\rm{m}}{\rm{c}}}})=max\left\{ \langle \widetilde{{\rho }_{{\rm{m}}{\rm{c}}}},W \rangle \,\Space{0ex}{0.25ex}{0ex}| 0\preceq W\preceq {\mathbb{1}},\,W\in \displaystyle \frac{1}{m}{{I}}^{\circ }\right\}.\end{eqnarray} \tag{ 80 }$$

Notice that ${W}^{\star }\in \tfrac{1}{m}{ \mathcal I }^\circ$ is equivalent to ${\max }_{i}{W}_{{ii}}\leqslant \tfrac{1}{m}$. Consider then the matrix ${W}_{\mathrm{mc}}:= {\sum }_{i,j}{W}_{{ij}}| {ii}\rangle \langle {jj}|$, defined in the basis of ${\rho }_{\mathrm{mc}}$. The eigenvalues of ${W}_{\mathrm{mc}}^{{T}_{B}}$ can be straightforwardly verified to be ${\{{W}_{{ii}},\,\pm | {W}_{{ij}}| \}}_{i,j=1}^{d}$. Positivity of ${W}^{\star }$ imposes that

$$\begin{eqnarray}&&\mathop{\max }\limits_{i,j}| {W}_{{ij}}| \leqslant \mathop{\max }\limits_{i}{W}_{{ii}}\end{eqnarray} \tag{ 81 }$$

from which it follows that

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{\mathsf{PPT}}{{\prime} }^{^\circ }}({W}_{\mathrm{mc}})={\parallel {W}_{\mathrm{mc}}^{{T}_{B}}\parallel }_{\infty }=\mathop{\max }\limits_{i}{W}_{{ii}}\leqslant \displaystyle \frac{1}{m}.\end{eqnarray} \tag{ 82 }$$

This means that ${W}_{\mathrm{mc}}\in \tfrac{1}{m}{\mathsf{PPT}}^{\prime}\circ$, and so

$$\begin{eqnarray}\begin{array}{rcl}{F}_{\mathrm{PPT}}({\rho }_{\mathrm{mc}},m) & = & {G}_{{\mathsf{PPT}}^{\prime} }^{(m)}({\rho }_{\mathrm{mc}})=\max \left\{\left\langle {\rho }_{\mathrm{mc}},W\right\rangle \left|0\preceq W\preceq {\mathbb{1}},\,W\in \displaystyle \frac{1}{m}{\mathsf{PPT}}^{\prime}\circ \right.\right\}\\ & \geqslant & \left\langle {\rho }_{\mathrm{mc}},{W}_{\mathrm{mc}}\right\rangle ={G}_{{ \mathcal I }}^{(m)}(\widetilde{{\rho }_{\mathrm{mc}}}).\end{array}\end{eqnarray} \tag{ 83 }$$

■

Notice that ${G}_{{ \mathcal I }}^{(m)}(\widetilde{{\rho }_{\mathrm{mc}}})=\tfrac{1}{m}\left({T}_{{ \mathcal I }}^{(m-1)}(\widetilde{{\rho }_{\mathrm{mc}}})+1\right)$, where ${T}_{{ \mathcal I }}^{(m-1)}$ have been considered as coherence measures in [61]. Further, using Theorem 5 we have that

$$\begin{eqnarray}&&{E}_{d,{ \mathcal O }}^{(1),\varepsilon }({\rho }_{\mathrm{mc}})={\left\lfloor \mathop{\min }\limits_{\sigma \in { \mathcal I }}{D}_{H}^{\varepsilon }(\widetilde{{\rho }_{\mathrm{mc}}}\parallel \sigma )\right\rfloor }_{\mathrm{log}}={\left\lfloor -\mathrm{log}\mathop{\min }\limits_{\displaystyle \genfrac{}{}{0em}{}{\left\langle \widetilde{{\rho }_{\mathrm{mc}}},W\right\rangle \geqslant 1-\varepsilon }{0\preceq W\preceq {\mathbb{1}}}}{\parallel {\rm{\Delta }}(W)\parallel }_{\infty }\right\rfloor }_{\mathrm{log}},\end{eqnarray} \tag{ 84 }$$

where ${\rm{\Delta }}(\cdot )={\sum }_{i}| i\rangle \langle i| \cdot | i\rangle \langle i|$ is the completely dephasing map. We stress that these optimisation problems are all efficiently computable as simple semidefinite programs [80], facilitating an efficient quantification of the fidelity as well as rates of one-shot distillation of all maximally correlated states.

Interestingly, in contrast to many other results which show an exact equality between operational quantities in the resource theory of coherence and the resource theory of entanglement of maximally correlated states, our result above shows a slight discrepancy between the two resources: in particular, it is not difficult to find numerical examples of states such that

$$\begin{eqnarray}&&{F}_{\mathrm{SEPP}}({\rho }_{\mathrm{mc}},m)={G}_{{ \mathcal I }}^{(m)}(\rho )\gt {G}_{{ \mathcal J }}^{(m)}(\rho )={F}_{\mathrm{MIO}}(\rho ,m)\end{eqnarray} \tag{ 85 }$$

$\forall m\lt d$, where MIO denotes the class of maximally incoherent operations in the resource theory of coherence, defined to be channels Λ such that $\sigma \in { \mathcal I }\Rightarrow {\rm{\Lambda }}(\sigma )\in { \mathcal I }$, and ${ \mathcal J }$ is the set of all unit-trace diagonal Hermitian operators (see [61] for the rightmost equality). Therefore, the one-shot distillable entanglement of a maximally correlated state under the largest set of free operations in the resource theory of entanglement (SEPP) can be strictly larger than the distillable coherence of the corresponding single-partite state under the largest set of free operations in the resource theory of coherence (MIO). This shows in particular that, in the distillation of entanglement from maximally correlated states under SEPP, it is not sufficient to consider operations whose output remains in the maximally correlated subspace—indeed, if this were the case, any such operation could always be mapped to a corresponding MIO operation, and the fidelities ${F}_{\mathrm{MIO}}(\rho ,m)$ and ${F}_{\mathrm{SEPP}}({\rho }_{\mathrm{mc}},m)$ would be equal. This also motivates a rather curious conjecture that, should there exist a smaller class of operations for which it suffices to consider only maximally correlated output states, then it is plausible that ${F}_{\mathrm{LOCC}}({\rho }_{\mathrm{mc}},m)\leqslant {F}_{\mathrm{MIO}}(\rho ,m)\lt {F}_{\mathrm{PPT}}({\rho }_{\mathrm{mc}},m)$ for general maximally correlated states. This could be surprising, as it is known that the gap between $\mathrm{LOCC}$ and $\mathrm{PPT}$ distillation of ${\rho }_{\mathrm{mc}}$ disappears at the asymptotic level [11, 81] or even when considering the second-order non-asymptotic expansion of the rate of distillation [20].

The setting of (environment-)assisted distillation of entanglement, considered first in [31, 32], has been studied in the non-asymptotic regime in [17]. It is based on a scenario in which the two parties A and B are assisted by a third party C who holds a purifying state of the system ${\rho }_{{AB}}$, i.e. such that the joint state is ${\psi }_{{ABC}}$, and aims to increase the entanglement distillable from ${\rho }_{{AB}}$ by performing a measurement on their local system C and communicating its result classically to A and B. A particular property of this setting is that the optimal protocol always involves a rank-1 measurement on subsystem C [17], giving parties A and B access to arbitrary pure-state decompositions of the system ${\rho }_{{AB}}$. Specifically, the best achievable rate of distillation is given by

$$\begin{eqnarray}&&{E}_{A,{ \mathcal O }}^{(1),\varepsilon }(\rho ):= \mathrm{log}\,\max \left\{\left.m\in {\mathbb{N}}\right|{F}_{A,{ \mathcal O }}(\rho ,m)\geqslant 1-\varepsilon \right\},\end{eqnarray} \tag{ 86 }$$

where the fidelity of assisted distillation is the best average fidelity optimised over all decompositions, i.e.

$$\begin{eqnarray}&&{F}_{A,{ \mathcal O }}(\rho ,m):= =\max \left.\left\{\left\langle \displaystyle \sum _{i}{p}_{i}{{\rm{\Lambda }}}_{i}(| {\psi }_{i}\rangle \langle {\psi }_{i}| ),{{\rm{\Psi }}}_{m}\right\rangle \,\right|\rho =\displaystyle \sum _{i}{p}_{i}| {\psi }_{i}\rangle \langle {\psi }_{i}| ,{{\rm{\Lambda }}}_{i}\in { \mathcal O }\,\forall i\right\}.\end{eqnarray} \tag{ 87 }$$

This in particular means that, having obtained the measurement result from party C, the distillation is performed from a pure state—therefore, employing our results in Theorem 15, we immediately obtain the result that the rate of assisted entanglement disitillation is the same under all sets of operations from $1\mbox{--}\mathrm{LOCC}$ up to ${\mathrm{PPTP}}_{+}$ and $\mathrm{SEPP}$.

Additionally, the proofs of the main results of [17] can be significantly simplified by employing the formalism introduced in this work, in fact strengthening the one-shot characterisation of Theorems 1 and 2 of [17] and tightening the bounds derived therein. In particular, our pure state-results in Theorem 15 allow us to straightforwardly obtain the following.

Theorem 20. For any ${ \mathcal O }\in \{1\mbox{--}\mathrm{LOCC}$, $\mathrm{LOCC}$, $\mathrm{PPT}$, $\mathrm{PPTP}$, ${\mathrm{PPTP}}_{+}$, $\mathrm{SEPP}\}$, the fidelity and one-shot rate of assisted distillation of any state are given by

$$\begin{eqnarray}&&{F}_{A,{ \mathcal O }}(\rho ,m)=\max \left\{F(\rho ,\omega )\left|\omega \in {{ \mathcal M }}_{m}\right.\right\},\end{eqnarray} \tag{ 88 }$$

$$\begin{eqnarray}&&{E}_{A,{ \mathcal O }}^{(1),\varepsilon }(\rho )={\left\lfloor -\mathrm{log}\min \left\{\left.\vartheta (\omega )\right|\omega \in {\mathbb{D}},F(\rho ,\omega )\geqslant 1-\varepsilon \right\}\right\rfloor }_{\mathrm{log}},\end{eqnarray} \tag{ 89 }$$

where

$$\begin{eqnarray}&&{{ \mathcal M }}_{m}:= \,\mathrm{conv}\left\{| \phi \rangle \langle \phi | \left|{\parallel | \phi \rangle \parallel }_{{{\ell }}_{2}}=1,\,{\parallel | {\xi }_{\phi }\rangle \parallel }_{{{\ell }}_{\infty }}\leqslant \displaystyle \frac{1}{\sqrt{m}}\right.\right\}\end{eqnarray} \tag{ 90 }$$

and

$$\begin{eqnarray}&&\vartheta (\omega ):= \min \left\{\left.\mathop{\max }\limits_{i}{\parallel | {\xi }_{{\psi }_{i}}\rangle \parallel }_{{{\ell }}_{\infty }}^{2}\right|\omega =\displaystyle \sum _{i}{p}_{i}| {\psi }_{i}\rangle \langle {\psi }_{i}| \right\}.\end{eqnarray} \tag{ 91 }$$

Proof. The derivation follows the approach taken for quantum coherence in [82]. We begin by writing the fidelity of assisted distillation as

$$\begin{eqnarray}&&{F}_{A,{ \mathcal O }}(\rho ,m)=\max \left\{\left.\displaystyle \sum _{i}{p}_{i}{F}_{{ \mathcal O }}({\psi }_{i},m)\right|\rho =\displaystyle \sum _{i}{p}_{i}| {\psi }_{i}\rangle \langle {\psi }_{i}| \right\}\end{eqnarray} \tag{ 92 }$$

with ${F}_{{ \mathcal O }}$ denoting the fidelity of distillation as before. Notice now that ${F}_{{ \mathcal O }}({\psi }_{i},m)=\tfrac{1}{m}{\parallel | {\xi }_{{\psi }_{i}}\rangle \parallel }_{\,[m]}^{2}$ can be written as ${F}_{{ \mathcal O }}({\psi }_{i},m)={\max }_{\omega \in {{ \mathcal M }}_{m}}F({\psi }_{i},\omega )$, where we employed the dual characterisation of the m-distillation norm of the Schmidt vector. Since ${{ \mathcal M }}_{m}$ is defined as the convex hull of rank-one projectors, we can now use the result of Streltsov et al [83] (see also [82]) to obtain

$$\begin{eqnarray}&&{F}_{A,{ \mathcal O }}(\rho ,m)=\max \left\{\left.\displaystyle \sum _{i}{p}_{i}\mathop{\max }\limits_{{\omega }_{i}\in {{ \mathcal M }}_{m}}F({\psi }_{i},{\omega }_{i})\right|\rho =\displaystyle \sum _{i}{p}_{i}| {\psi }_{i}\rangle \langle {\psi }_{i}| \right\}=\mathop{\max }\limits_{\omega \in {{ \mathcal M }}_{m}}F(\rho ,\omega )\end{eqnarray} \tag{ 93 }$$

as required. The quantity ϑ is simply a function defined so that any state ω satisfies $\omega \in {{ \mathcal M }}_{m}\ \Longleftrightarrow \ \vartheta (\omega )\leqslant \tfrac{1}{m}$, allowing us to obtain

$$\begin{eqnarray}\begin{array}{rcl}{E}_{A,{ \mathcal O }}^{(1),\varepsilon }(\rho ) & := & \mathrm{log}\,\max \left\{m\in {\mathbb{N}}\left|\mathop{\max }\limits_{\omega \in {{ \mathcal M }}_{m}}F(\rho ,\omega )\geqslant 1-\varepsilon \right.\right\}\\ & = & \mathrm{log}\,\max \left\{m\in {\mathbb{N}}\left|\,\omega \in {\mathbb{D}},\,\vartheta (\omega )\leqslant \displaystyle \frac{1}{m},\right.\,F(\rho ,\omega )\geqslant 1-\varepsilon \right\}\end{array}\end{eqnarray} \tag{ 94 }$$

and thus completing the proof. ■

Taking $\varepsilon =0$ in the task of one-shot distillation corresponds to the problem of characterising the exact transformation $\rho \to {{\rm{\Psi }}}_{m}$ with a given class of free operations. One is then interested in understanding not only the one-shot capabilities in such a task, but also the asymptotically achievable rate

$$\begin{eqnarray}&&{E}_{d,{ \mathcal O }}^{\infty ,0}(\rho ):= \mathop{\mathrm{lim}\,\sup }\limits_{n\to \infty }\displaystyle \frac{1}{n}{E}_{d,{ \mathcal O }}^{(1),0}({\rho }^{\otimes n}).\end{eqnarray} \tag{ 95 }$$

To apply our methods in this setting, let us focus on the classes of operations for which we have shown that ${F}_{{ \mathcal O }}(\rho ,m)={G}_{{ \mathcal S }}^{(m)}(\rho )$ for some set ${ \mathcal S };$ recall from our previous results that ${F}_{\mathrm{SEPP}}(\cdot ,m)={G}_{{\mathsf{SEP}}}^{(m)}$, ${F}_{{\mathrm{PPTP}}_{+}}(\cdot ,m)={G}_{{{\mathsf{PPT}}}_{+}}^{(m)}$, ${F}_{{\mathrm{PPTP}}_{+}^{\prime }}={G}_{{{\mathsf{PPT}}}_{+}^{{\prime} }}^{(m)}$, and ${F}_{\mathrm{PPTP}}(\cdot ,m)={F}_{\mathrm{PPT}}(\cdot ,m)={G}_{{\mathsf{PPT}}^{\prime} }^{(m)}$.

Lemma 21. Take ${ \mathcal O }\in \{\mathrm{PPT},\mathrm{PPTP},{\mathrm{PPTP}}_{+}^{\,^{\prime} },{\mathrm{PPTP}}_{+},\mathrm{SEPP}\}$ and let ${ \mathcal S }$ be the set such that ${F}_{{ \mathcal O }}(\rho ,m)={G}_{{ \mathcal S }}^{(m)}(\rho )$ for the given class. Then, for any ρ, it holds that

$$\begin{eqnarray}&&{E}_{d,{ \mathcal O }}^{(1),0}(\rho )={\left\lfloor -\mathrm{log}\min \left\{\left.{{\rm{\Gamma }}}_{{{ \mathcal S }}^{^\circ }}(W)\right|{{\rm{\Pi }}}_{\rho }\preceq W\preceq {\mathbb{1}}\right\}\right\rfloor }_{\mathrm{log}},\end{eqnarray} \tag{ 96 }$$

where ${{\rm{\Pi }}}_{\rho }$ is the projector onto the support of ρ.

Proof. Using the characterisation in Proposition 4 and Theorem 5, we can write

$$\begin{eqnarray}&&{E}_{d,{ \mathcal O }}^{(1),0}(\rho )={\left\lfloor -\mathrm{log}\min \left\{\left.{{\rm{\Gamma }}}_{{{ \mathcal S }}^{^\circ }}(W)\right|\left\langle \rho ,W\right\rangle =1,0\preceq W\preceq {\mathbb{1}}\right\}\right\rfloor }_{\mathrm{log}}.\end{eqnarray} \tag{ 97 }$$

Write ρ in its spectral decomposition as $\rho ={\sum }_{i}{\lambda }_{i}| {\psi }_{i}\rangle \langle {\psi }_{i}|$. We then have

$$\begin{eqnarray}&&\displaystyle \sum _{i}{\lambda }_{i}=\mathrm{Tr}\rho =1=\left\langle W,\rho \right\rangle =\displaystyle \sum _{i}{\lambda }_{i}\left\langle W,| {\psi }_{i}\rangle \langle {\psi }_{i}| \right\rangle \end{eqnarray} \tag{ 98 }$$

and so $\left\langle {\psi }_{i}| W| {\psi }_{i}\right\rangle =1$ for each $i\in \{1,\ldots ,r\}$ since $0\preceq W\preceq {\mathbb{1}}$. The constraints then imply that every feasible solution will have the form $W={{\rm{\Pi }}}_{\rho }+P$ with $0\preceq P\preceq {\mathbb{1}}$ and $\mathrm{supp}(P)\subseteq \ker (\rho )$, and in particular ${{\rm{\Pi }}}_{\rho }\preceq W\preceq {\mathbb{1}}$. Conversely, every W such that ${{\rm{\Pi }}}_{\rho }\preceq W\preceq {\mathbb{1}}$ satisfies $1\geqslant \left\langle \rho ,W\right\rangle \geqslant 1$ and $0\preceq W\preceq {\mathbb{1}}$, so the feasible sets of the two problems are equal. ■

Note that for any positive semidefinite W, from equation (10) we have ${{\rm{\Gamma }}}_{{{ \mathcal S }}^{^\circ }}(W)={\max }_{X\in { \mathcal S }}\left\langle X,W\right\rangle$. For the case of $\mathrm{PPT}$ operations, where ${{\rm{\Gamma }}}_{{{\mathsf{PPT}}}^{{\prime} ^\circ }}(W)={\parallel {W}^{{T}_{B}}\parallel }_{\infty }$, the above recovers a result of [84].

Notice that the above implies that one-shot zero-error distillation is impossible from any full-rank state under any class of free operations, as for ${{\rm{\Pi }}}_{\rho }={\mathbb{1}}$ the only feasible W is ${\mathbb{1}}$ itself and so we have ${E}_{d,{ \mathcal O }}^{(1),0}(\rho )=\mathrm{log}\left\lfloor {{\rm{\Gamma }}}_{{{ \mathcal S }}^{^\circ }}{\left({\mathbb{1}}\right)}^{-1}\right\rfloor =\mathrm{log}1=0$. We will shortly improve this characterisation of zero-error undistillability.

Interestingly, in the case of ${\mathrm{PPTP}}_{+}$, ${\mathrm{PPTP}}_{+}^{\,^{\prime} }$, and $\mathrm{SEPP}$, the set ${ \mathcal S }$ consists of positive semidefinite operators, which means that for any $P\in {{\mathbb{H}}}_{+}$ it holds that

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{{ \mathcal S }}^{^\circ }}({{\rm{\Pi }}}_{\rho }+P)=\mathop{\max }\limits_{\sigma \in { \mathcal S }}\left\langle \sigma ,{{\rm{\Pi }}}_{\rho }+P\right\rangle \geqslant \mathop{\max }\limits_{\sigma \in { \mathcal S }}\left\langle \sigma ,{{\rm{\Pi }}}_{\rho }\right\rangle ={{\rm{\Gamma }}}_{{{ \mathcal S }}^{^\circ }}({{\rm{\Pi }}}_{\rho })\end{eqnarray} \tag{ 99 }$$

and so ${{\rm{\Pi }}}_{\rho }$ itself will be the optimal solution to the minimisation in equation (96). This gives the following.

Corollary 22. For the classes of operations ${ \mathcal O }\in \{{\mathrm{PPTP}}_{+},{\mathrm{PPTP}}_{+}^{\,^{\prime} },\mathrm{SEPP}\}$, the one-shot zero-error distillable entanglement is given exactly by

$$\begin{eqnarray}&&{E}_{d,{ \mathcal O }}^{(1),0}(\rho )=\mathrm{log}\left\lfloor {{\rm{\Gamma }}}_{{{ \mathcal S }}^{^\circ }}{\left({{\rm{\Pi }}}_{\rho }\right)}^{-1}\right\rfloor .\end{eqnarray} \tag{ 100 }$$

Noticing further that ${{\rm{\Pi }}}_{{\rho }^{\otimes n}}={{\rm{\Pi }}}_{\rho }^{\otimes n}$, we can easily see that ${{\rm{\Gamma }}}_{{{ \mathcal S }}^{^\circ }}\left({{\rm{\Pi }}}_{{\rho }^{\otimes n}}\right)\geqslant {{\rm{\Gamma }}}_{{{ \mathcal S }}^{^\circ }}{\left({{\rm{\Pi }}}_{\rho }\right)}^{n}$ due to the fact that $\sigma \in { \mathcal S }\Rightarrow {\sigma }^{\otimes n}\in { \mathcal S }$. This gives the relation ${E}_{d,{ \mathcal O }}^{(1),0}({\rho }^{\otimes n})\leqslant {\left\lfloor -n\mathrm{log}{{\rm{\Gamma }}}_{{{ \mathcal S }}^{^\circ }}({{\rm{\Pi }}}_{\rho })\right\rfloor }_{\mathrm{log}}$, and in particular we see that $-\mathrm{log}{{\rm{\Gamma }}}_{{{ \mathcal S }}^{^\circ }}({{\rm{\Pi }}}_{\rho })$ upper bounds the asymptotically achievable zero-error distillable entanglement ${E}_{d,{ \mathcal O }}^{\infty ,0}(\rho )$. Equality does not generally hold since the quantities ${{\rm{\Gamma }}}_{{{\mathsf{SEP}}}^{^\circ }}$, ${{\rm{\Gamma }}}_{{{\mathsf{PPT}}}_{+}^{^\circ }}$ are not multiplicative (a counterexample being any state supported on the antisymmetric subspace [85, 86]). Interestingly, multiplicativity is indeed satisfied for ${{\rm{\Gamma }}}_{{{{\mathsf{PPT}}}_{+}^{{\prime} }}^{^\circ }}$—this can be seen explicitly by expressing the computation of ${{\rm{\Gamma }}}_{{{{\mathsf{PPT}}}_{+}^{{\prime} }}^{^\circ }}$ in its dual form as

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{{{\mathsf{PPT}}}_{+}^{{\prime} }}^{^\circ }}({{\rm{\Pi }}}_{\rho })=\mathop{\min }\limits_{Q\succcurlyeq {{\rm{\Pi }}}_{\rho }}{\parallel {Q}^{{T}_{B}}\parallel }_{\infty },\end{eqnarray} \tag{ 101 }$$

from which it straightforwardly follows that ${{\rm{\Gamma }}}_{{{{\mathsf{PPT}}}_{+}^{{\prime} }}^{^\circ }}({{\rm{\Pi }}}_{\rho }^{\otimes n})\leqslant {{\rm{\Gamma }}}_{{{{\mathsf{PPT}}}_{+}^{{\prime} }}^{^\circ }}{\left({{\rm{\Pi }}}_{\rho }\right)}^{n}$. This gives in particular the following.

Corollary 23. The asymptotic zero-error distillable entanglement under Rains-preserving operations is given by

$$\begin{eqnarray}&&{E}_{d,{ \mathcal O }}^{\infty ,0}(\rho )=-\mathrm{log}{{\rm{\Gamma }}}_{{{{\mathsf{PPT}}}_{+}^{{\prime} }}^{^\circ }}({{\rm{\Pi }}}_{\rho }).\end{eqnarray} \tag{ 102 }$$

The result therefore ensures the computability of both one-shot and asymptotic zero-error distillable entanglement under ${\mathrm{PPTP}}_{+}^{\,^{\prime} }$, showing that it constitutes an efficiently computable upper bound for zero-error LOCC distillation. Note that ${{\rm{\Gamma }}}_{{{{\mathsf{PPT}}}_{+}^{{\prime} }}^{^\circ }}({{\rm{\Pi }}}_{\rho })$ appeared previously in the works [33, 58] as a bound on entanglement cost and zero-error distillable entanglement. Our result gives this quantity a precise operational meaning, establishing it as a zero-error equivalent of the Rains bound (see section 3.1.1; see also discussion in [33]).

Evaluating ${{\rm{\Gamma }}}_{{{\mathsf{SEP}}}^{^\circ }}$ is significantly more difficult [87, 88]. One can write this quantity more explicitly as [89]

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{{\mathsf{SEP}}}^{^\circ }}({{\rm{\Pi }}}_{\rho })=\max \left\{\left.{\parallel | {\xi }_{\psi }\rangle \parallel }_{\,{{\ell }}_{\infty }}^{2}\right|| \psi \rangle \in \mathrm{supp}(\rho )\right\}\end{eqnarray} \tag{ 103 }$$

which makes it easy to see that if the support of ρ contains a product state, then no class of free operations can distil any entanglement without error (even asymptotically). By a result of Parthasarathy [90], if $\mathrm{rank}(\rho )\gt ({d}_{A}-1)({d}_{B}-1)$, then ${{\rm{\Gamma }}}_{{{\mathsf{SEP}}}^{^\circ }}({{\rm{\Pi }}}_{\rho })=1$ and so ${E}_{d,\mathrm{SEPP}}^{(1),0}(\rho )=0$. In a very similar manner, if $\mathrm{supp}(\rho )$ contains a PPT state, then ${E}_{d,{\mathrm{PPTP}}_{+}}^{(1),0}(\rho )=0;$ this, however, does not give a better universal bound for the rank of ρ which ensures undistillability [91].

Our results in previous sections can further simplify the characterisation of zero-error distillable entanglement for several classes of states. In particular, any pure state has

$$\begin{eqnarray}&&{E}_{d,{ \mathcal O }}^{(1),0}(\psi )=\mathrm{log}\left\lfloor {\parallel | {\xi }_{\psi }\rangle \parallel }_{\,{{\ell }}_{\infty }}^{-2}\right\rfloor \end{eqnarray} \tag{ 104 }$$

for any class of operations ${ \mathcal O }$ considered in this work (which was already known in the case of LOCC [12] and PPT operations [52]), and a maximally correlated state satisfies

$$\begin{eqnarray}&&{E}_{d,{ \mathcal O }}^{(1),0}({\rho }_{\mathrm{mc}})=\mathrm{log}\left\lfloor {\parallel {\rm{\Delta }}({{\rm{\Pi }}}_{{\rho }_{\mathrm{mc}}})\parallel }_{\,\infty }^{-1}\right\rfloor \end{eqnarray} \tag{ 105 }$$

for any ${ \mathcal O }\in \{\mathrm{PPT},{\mathrm{PPTP}}_{+}^{\,^{\prime} },{\mathrm{PPTP}}_{+},\mathrm{SEPP}\}$, where Δ is the completely dephasing channel (diagonal map) in the maximally correlated basis. One can furthermore notice that in both of the above cases the quantity ${{\rm{\Gamma }}}_{{{\mathsf{SEP}}}^{^\circ }}({{\rm{\Pi }}}_{\rho })$ is multiplicative, which means that in the asymptotic limit we have ${E}_{d,{ \mathcal O }}^{\infty ,0}(\psi )=-\mathrm{log}{\parallel | {\xi }_{\psi }\rangle \parallel }_{\,{{\ell }}_{\infty }}^{2}$ and ${E}_{d,{ \mathcal O }}^{\infty ,0}({\rho }_{\mathrm{mc}})=-\mathrm{log}{\parallel {\rm{\Delta }}({{\rm{\Pi }}}_{{\rho }_{\mathrm{mc}}})\parallel }_{\infty }$.

Finally, we remark that the quantity ${{\rm{\Gamma }}}_{{{\mathsf{SEP}}}^{^\circ }}$, often encountered under the name ${h}_{{\mathsf{SEP}}}$, has found a plethora of uses beyond the resource theory of entanglement—in particular, in the theory of quantum Merlin–Arthur games [92] as well as in characterising the maximum output norms of quantum channels [85, 92]. Indeed, the non-multiplicativity of ${{\rm{\Gamma }}}_{{{\mathsf{SEP}}}^{^\circ }}$ is equivalent to the non-multiplicativity of the norm ${\parallel {\rm{\Lambda }}\parallel }_{1\to \infty }:= \max \left\{\left.{\parallel {\rm{\Lambda }}(\rho )\parallel }_{\infty }\right|\rho \in {\mathbb{D}}\right\}$ of a channel Λ; specifically, if Λ takes operators on a Hilbert space ${{ \mathcal H }}_{\mathrm{in}}$ to operators on Hilbert space ${{ \mathcal H }}_{\mathrm{out}}$ and $V:{{ \mathcal H }}_{\mathrm{in}}\to {{ \mathcal H }}_{\mathrm{out}}\otimes {{ \mathcal H }}_{R}$ is an isometry such that ${\rm{\Lambda }}(\cdot )={\mathrm{Tr}}_{R}V\cdot {V}^{\dagger }$ for some auxiliary Hilbert space ${{ \mathcal H }}_{R}$, then ${\parallel {\rm{\Lambda }}\parallel }_{1\to \infty }={{\rm{\Gamma }}}_{{{\mathsf{SEP}}}^{^\circ }}({{VV}}^{\dagger })$ [92]. This interpretation provides an understanding of the cases in which ${{\rm{\Gamma }}}_{{{\mathsf{SEP}}}^{^\circ }}({{VV}}^{\dagger })$ is multiplicative: these are the cases in which the protocol Λ obeys so-called perfect parallel repetition [92]. It is furthermore known that, although not multiplicative, the quantity ${{\rm{\Gamma }}}_{{{\mathsf{SEP}}}^{^\circ }}$ obeys a form of weaker multiplicativity relations [93, 94]. In the context of entanglement distillation we can see that additivity, in the sense that ${E}_{d,\mathrm{SEPP}}^{\infty ,0}(\rho )=-\mathrm{log}{{\rm{\Gamma }}}_{{{\mathsf{SEP}}}^{^\circ }}({{\rm{\Pi }}}_{\rho })$, holds when the optimal operation ${\rm{\Lambda }}\in \mathrm{SEPP}$ which distils entanglement from ρ satisfies ${{\rm{\Lambda }}}^{\otimes n}\in \mathrm{SEPP}$ for any n. We stress that an additive lower bound on ${E}_{d,\mathrm{SEPP}}^{\infty ,0}(\rho )$, and therefore also an upper bound on the regularisation of ${\parallel \cdot \parallel }_{1\to \infty }$, is given by Corollary 23.