Entanglement manipulation beyond local operations and classical communication

In the distant laboratory paradigm of quantum information theory, a multipartite quantum system is distributed to spatially separated parties. The individual parties are restricted to performing only local quantum operations on their respective subsystems, but they are permitted to coordinate their actions through global classical communication. Quantum channels arising from this scenario are known as LOCC (local operations and classical communication), and they emerge as the natural operational class to consider for distributed quantum information processing tasks. Entanglement becomes a resource under the restriction of LOCC; it is consumed when performing information processing tasks (such as quantum teleportation), and entangled states cannot be generated using LOCC alone. It is, therefore, natural to study entanglement in the framework of quantum resource theory, which characterizes the convertibility of entangled states under LOCC and investigates their applicability in various information-theoretic tasks (see Refs. 1–3 for review).

Despite the conceptually intuitive description of LOCC in the distant-lab setting, mathematically characterizing the set of LOCC operations is notoriously complex.⁴ One way to overcome this difficulty is by relaxing the constraints of LOCC to encompass a more mathematically simple class of operations. Any task found to be impossible by these more general operations is, therefore, also impossible by LOCC. In fact, LOCC is not the largest class of channels that cannot create entanglement from unentangled, or separable, states. There exists a plethora of operational classes “beyond” LOCC that act invariantly on the set of separable states, and each of these defines a different quantum resource theory of entanglement manipulation.

It is interesting to study what differences arise in these generalized resource theories of entanglement compared to the standard LOCC scenario. Doing so has already proven to be successful in the asymptotic scenario where, under LOCC, certain mixed states demonstrate irreversibility in terms of pure-state entanglement distillation and the reverse process of entanglement dilution.^5,6 The most dramatic example of this irreversibility is the phenomenon of bound entanglement,⁷ which refers to entangled mixed states that require the consumption of pure-state entanglement for their preparation, and yet no pure-state entanglement can be distilled back from them. Remarkably, the theory of entanglement distillation and dilution becomes much more amenable and elegant after enlarging the class of operations to include all asymptotically non-entangling operations. The work in Refs. 8–10 shows that bound entanglement no longer exists under this larger class of operations, and more importantly, the entanglement in any mixed state can be reversibly distilled and diluted at a rate given by the regularized relative entropy of entanglement.

In this paper, we analyze in detail the entanglement manipulation using operations beyond LOCC. Specifically, we study which state transformations are possible and which entanglement measures retain their validity within generalized resource theories of entanglement. For example, separable maps are those whose Choi representations are separable,¹¹ while non-entangling maps are precisely those that do not produce an entangled output state for any separable input. The class of dually non-entangling maps, which we introduce here, consists of channels Λ such that both Λ and its dual map Λ* are non-entangling. All classes of operations that we consider contain LOCC as a proper subset.

One of our motivations is to better understand the nature of bound entanglement within the standard framework of LOCC. While it is known that no pure-state entanglement can be distilled from any state having a positive partial transpose (PPT), one of the major open problems in quantum information theory is to determine if the converse is also true;¹² namely, does there exist non-PPT (NPT) bound entanglement? A strategy that would allow us to answer this question in the affirmative would be to identify a class of operations that contains LOCC, which lacks the ability to distill certain NPT states.

The first comprehensive attempt in this direction was carried out in Ref. 13, where it was shown that NPT bound entanglement does not exist when the class of operations is enlarged to contain all so-called PPT operations, as originally introduced in Refs. 14 and 15. This is the class of channels whose Choi representation is PPT (see also Sec. II for definitions). Even stronger, the results of Refs. 8–10 imply that no bound entanglement (not even PPT bound entanglement) exists for the class of all strictly non-entangling channels.

Beyond the question of distillability, we also examine the difference between the class of channels whose Choi operator is PPT and the class of PPT-preserving channels. While the former refers to the well-known class of PPT channels originally defined by Rains,^14,15 to our knowledge, the latter has not been previously studied. Finally, analogous to the relation between completely positive (CP) and k-positive maps, we also introduce here the classes of k-non-entangling and completely non-entangling maps. The former refers to maps that are non-entangling when tensored with the identity on a k-dimensional ancilla, while the latter refers to maps that are k-non-entangling for all k.

The rest of the paper is structured as follows. Section II introduces the necessary background and notation, including precise definitions for the classes of channels that we study. Section III investigates how certain entanglement measures behave under different classes of non-entangling channels. The entanglement measures studied in this section include the robustness of entanglement, the Schmidt rank, the Rényi α-entropies of entanglement (as well as the relative Rényi entropies of entanglement), and the negativity. Although the robustness remains monotonic under non-entangling channels, we use the robustness to derive conditions for the convertibility of pure states under non-entangling channels that are independent of the celebrated majorization criterion for LOCC channels.¹⁶ We also show that the Rényi α-entropies of entanglement (which are entanglement measures for all α ∈ [0, +∞]) can be increased arbitrarily under non-entangling channels for α ∈ [0, 1/2). On the other hand, we show that the α-entropies of entanglement in the range α ∈ [1/2, +∞] coincide with a related relative Rényi entropy of entanglement measure, and therefore, they cannot increase under non-entangling channels. In particular, this gives a closed-form expression for the relative α-entropies of entanglement in certain cases. We also demonstrate that the Schmidt rank can be increased under the more restrictive class of dually non-entangling maps. Finally, we show that the negativity can be increased by an arbitrarily large fractional amount under PPT-preserving maps, thus highlighting the difference between the classes of PPT and PPT-preserving maps.

Entanglement distillation by classes of channels larger than LOCC is investigated in Sec. IV. We demonstrate that all entangled states can be converted to an LOCC-distillable state using dually non-entangling channels. The significance of this result is somewhat tempered by the fact that two copies of a non-entangling channel need not be non-entangling, something that we will also demonstrate in Sec. IV. Consequently, single-copy convertibility of a given state to an LOCC-distillable one by some generalized class of operations does not imply that the original state is distillable within generalized resource theory. Finally, we close the section by showing that any class of operations that generalizes LOCC by placing a finite number of linear constraints on the Choi matrix always allows us to obtain LOCC-distillable states from any quantum state.

In Sec. V, we relax the constraint of trace-preserving (TP) maps and turn to the problem of stochastic convertibility between pure states. For multipartite state spaces, a natural way to categorize entanglement is in terms of stochastic convertibility under LOCC. That is, two states |ψ⟩ and $|ψ̂〉$ belong to the same entanglement class if there is invertible LOCC transformation from |ψ⟩ to $|ψ̂〉$ that succeeds with some nonzero probability.¹⁷ We show that this entanglement structure completely collapses under any class of entangling undetected channels. Connections to the problem of tensor rank calculation are also discussed.

The structure of k-non-entangling maps and k-PPT-preserving maps are analyzed in Sec. VI. We show that complete non-entangling is equivalent to d-non-entangling when the local systems have dimension d, and we show that the structure of k-non-entangling maps is nontrivial. That is, for every k < d, there exists k-non-entangling maps that are not (k + 1)-non-entangling. We also investigate distillability under this class of maps, but we have not been able to answer whether or not they allow for bound entanglement. In particular, we show that previously used constructions are not 3-non-entangling, so the weaker question of whether every entangled state can be converted to an LOCC-distillable one under this class of maps remains open. Finally, a detailed Proof of Theorem 4 is presented in  Appendix A, while a concluding discussion is given in Sec. VII.

Let $L(H)$ denote the space of linear operators on a finite-dimensional Hilbert space $H$ with orthonormal basis denoted by {|1⟩, …, |n⟩} (where n is the dimension of the space). An operator $ρ∈L(H)$ is in the set $D(H)$ of density operators (or states) on $H$ if ρ ≥ 0 and Tr(ρ) = 1. The tensor product of two Hilbert spaces $HA$ and $HB$ is denoted by $HA⊗HB$ or $HAB$⁠. The identity operator on a Hilbert space $HA$ is denoted as $1A$⁠, while the identity map on $L(HA)$ is denoted as id_A, and the identity operator on $Cd⊗Cd$ is denoted by $1d2$⁠.

We write ψ to denote the density operator ψ = |ψ⟩ ⟨ψ| associated with a pure state vector $|ψ〉∈H$⁠. For bipartite pure states $|ψ〉∈HAB$⁠, we typically assume, without loss of generality, that |ψ⟩ is in the Schmidt form

where |ii⟩ is the shorthand for |i⟩ ⊗ |i⟩ and λ = (λ₁, …, λ_d) are the Schmidt coefficients of ψ (in decreasing order) such that $∑i=1dλi=1$ and $d≤mindimHA,dimHB$⁠. The Schmidt rank of the state is defined to be the number of non-zero entries in λ, which we denote by rank(λ). A positive operator $σ∈L(HAB)$ is said to be separable with respect to A : B if it can be written as σ = ∑_iτ_i ⊗ ω_i for some positive operators $τi∈L(HA)$ and $ωi∈L(HB)$⁠. The set of separable density operators on $HAB$ will be denoted as $S(HA:B)$⁠, or simply $S$ if the spaces A and B are clear from context.

The partial transpose of an operator $X∈L(HAB)$⁠, denoted as $XΓA$⁠, is the operator that results from applying the transposition map A ↦ A^T (with respect to a given basis of $HA$⁠) to system A. We simply write X^Γ if the system being transposed is clear from context. A positive operator $ρ∈L(HAB)$ is positive under partial transpose (PPT) with respect to A : B if ρ^Γ ≥ 0, and it is otherwise said to be non-PPT (or NPT) (we note that the definition of PPT is independent of the basis chosen for the respective transposition). It is well-known that every separable state is PPT, with the converse holding true only in dimensions 2 ⊗ 2 and 2 ⊗ 3.¹⁸ 

For any integer d ≥ 2, the maximally entangled pure state on $Cd⊗Cd$ is denoted as

and we write $ϕd+=|ϕd+〉〈ϕd+|$ to denote the corresponding density operator. We denote the so-called “flip” operator on $Cd⊗Cd$ by

which is precisely the partial transpose of $ϕd+$ scaled by d. Analogously, we write $ϕA+$ to denote the (non-normalized) maximally entangled operator on the Hilbert space $HA⊗HA$⁠,

where {|i⟩} is an orthonormal basis for $HA$⁠. Note that we can write the maximally entangled operator of the four-partite space $ϕAB+∈L(HAB⊗HAB)$ as

Given a linear map $Λ:L(HA1)→L(HA2)$⁠, its Choi representation is the linear operator $J(Λ)∈L(HA2A1)$ defined by

For any $X∈L(HA1)$⁠, the action of the map Λ on X can be written in terms of its Choi representation as $Λ(X)=TrA1(J(Λ)(1A2⊗XT))$⁠. Recall that Λ is completely positive (CP) if and only if its Choi representation is a positive semi-definite operator, and Λ is trace preserving (TP) if and only if it holds that $TrA2(J(Λ))=1A1$⁠. A linear map that is both completely positive and trace preserving (CPTP) is called a quantum channel. Given two density operators $ρ1,ρ2∈D(H′)$ and an operator $A∈L(H)$⁠, the linear map $Λ:L(H)→L(H′)$ defined by

is CPTP as long as 0 ≤ A ≤ $1A$⁠. Many of the maps constructed in this work take this form. Given a linear map $Λ:L(H)→L(H′)$⁠, its dual map $Λ*:L(H′)→L(H)$ is the unique linear map such that Tr(Λ(X)Y) = Tr(XΛ*(Y)) holds for all operators $X∈L(H)$ and $Y∈L(H′)$⁠. It follows readily from the definition that the dual of a map of the form in (1) is given by

while the Choi representation of a map of the form in (1) is given by

We now define all the classes of operations that will be explored in this paper. We first recall the definitions of separable and PPT maps. Let $Λ:L(HA1B1)→L(HA2B2)$ be a completely positive linear map. Then, Λ is said to be separable if its Choi representation $J(Λ)∈L(HA2B2⊗HA1B1)$ is separable with respect to A₂A₁ : B₂B₁. To keep the bipartition clear, we will often denote $HA2B2⊗HA1B1$ as $HA2A1:B2B1$⁠. Note that Λ is separable if and only if it can be written as

for some CP maps $Φj:L(HA1)→L(HA2)$ and $Ψj:L(HB1)→L(HB2)$⁠. Analogously, the map Λ is a PPT map if the partial transpose of its Choi representation is a positive semi-definite, i.e., if $J(Λ)Γ=J(Λ)ΓA2A1≥0$⁠. It holds that Λ is PPT if and only if the map Λ^Γ defined by

is also completely positive. In particular, note that J(Λ^Γ) = J(Λ)^Γ. This definition of PPT maps is due to Rains.¹⁴ 

In this paper, we are primarily concerned with the non-entangling maps, which are defined explicitly as follows. We provide here an analogous definition for PPT-preserving maps.

By definition, the non-entangling channels comprise the largest class of channels which cannot create entanglement, while the PPT-preserving channels are exactly those that cannot generate an NPT state from PPT input states. It is clear that every separable (respectively PPT) map is separable-preserving (respectively PPT-preserving). We remark that the classes of separable-preserving and PPT-preserving maps are distinct from the classes separable and PPT maps. Indeed, the bipartite swap operator is clearly both separable- and PPT-preserving, but its corresponding Choi representation is proportional to the maximally entangled state which is neither separable nor PPT.

We will also consider the following set of maps that is even more restricted than the non-entangling maps which nonetheless contains the set of LOCC channels.

Note that a map is completely positive if and only if its dual map is, but the dual of a trace-preserving map is not necessarily trace-preserving. Dually non-entangling channels form a subset of the non-entangling channels which still contains all LOCC. Indeed, the dual of any separable map is clearly separable; hence, separable maps are dually non-entangling (and thus, LOCC channels are as well).

Any quantum channel is a linear map that must not only preserve positivity but also preserve positivity when it acts on the systems tensored with any ancilla space for a linear map to represent a physical manipulation of quantum states. Motivated by this requirement of complete positivity for a quantum channel, in the following, we define a notion of complete non-entangling. The analogous definition is provided for PPT-preserving maps. We note that completely non-entangling maps (respectively, completely PPT-preserving maps) are exactly those maps that are separable (respectively, PPT). This will be discussed and proved in Sec. VI.

For the final family of channels that we will use, we recall the notion of entanglement witnesses. A Hermitian operator $W∈L(HAB)$ is an entanglement witness with respect to A : B if it holds that Tr(Wρ) ≥ 0 for all separable states $ρ∈S(HA:B)$⁠. Since the Choi operator for any separable map is separable, we can use entanglement witnesses on the level of Choi operators to detect non-separable maps. Even stronger, it is known that entanglement witnesses provide a full characterization of separability in the following sense. A positive operator X is separable if and only if Tr(WX) ≥ 0 holds for all entanglement witnesses W.¹⁸ 

Geometrically, every entanglement witness can be seen as a hyperplane in the space of Hermitian operators that separates the positive convex cone of separable operators (see Fig. 1). Furthermore, the set of separable operators is not a polytope,^19,20 and therefore, any finite set of entanglement witnesses is insufficient to determine separability. Therefore, one obtains a strict relaxation of the separability constraint by requiring that Tr(WX) ≥ 0 hold for not all witnesses but instead for just some finite subset of witnesses. This idea motivates the following type of operations.

Note that for any convex set in $Rd$⁠, a containing polytope can be constructed that approximates the set to arbitrary precision.²¹ Thus, for any ϵ > 0, there exists a collection of entanglement witnesses ${Wi}i=1n$ such that a non-separable Λ is entangling undetected by ${Wi}i=1n$ only if ∥J(Λ) − Ω∥₁ < ϵ for all operators $Ω∈L(HA2B1A1B1)$ that are separable with respect to A₂A₁ : B₂B₁, where ∥·∥₁ denotes the trace norm.

In this section, we explore what kinds of transformations among states are possible under non-entangling channels but not possible under LOCC. This is done by finding transformations under non-entangling channels that increase some entanglement measure. We examine which entanglement measures retain their monotonicity under non-entangling channels, dually non-entangling channels, and PPT-preserving channels.

We first explore certain entanglement measures that can be increased with non-entangling channels and show how certain other measures can yield conditions for conversion under such maps. Although most of the claims in this section follow from the already known results, it is useful for our purposes to collect information about which entanglement measures remain monotonic under all non-entangling maps and which ones do not.

Consider first the relative entropy of entanglement,^22,23

where S(ρ‖σ) ≔ Tr(ρ log ρ) − Tr(ρ log σ) is the relative entropy of ρ and σ. This is an entanglement measure that is clearly also monotonic under non-entangling channels, a fact that follows directly from the joint monotonicity of the relative entropy, i.e., S(Λ(ρ)‖Λ(σ)) ≤ S(ρ‖σ) for every CPTP map Λ.²⁴ Another entanglement measure that retains its monotonicity under non-entangling channels, as already observed in Ref. 9, is the robustness of entanglement,²⁵ which is defined as

where, for any separable state $σ∈S$⁠,

Indeed, if Λ is a non-entangling channel, it is clearly seen that R(Λ(ρ)‖Λ(σ)) ≤ R(ρ‖σ) holds for any separable σ. For pure states, the robustness reduces (see Ref. 25) to

In particular, it holds that $R(ϕk+)=k−1$ for any integer k ≥ 2.

The robustness yields necessary and sufficient conditions for certain transformations among certain states under non-entangling channels. The following result provides necessary and sufficient conditions for the transformation of a maximally entangled state to any other state purely in terms of the robustness. This is effectively a statement about the exact one-shot entanglement cost under non-entangling channels and, thus, follows straightforwardly from Theorem 2 in Ref. 10 (similar methods are also used in Ref. 9). For the sake of completeness, we provide a proof of this fact as it presents techniques and ideas that will be used later in this work.

Whereas pure state transformation under LOCC is governed strictly by majorization of Schmidt coefficients,¹⁶ the result of Theorem 1 indicates that in certain cases, the Schmidt rank of a pure state (which cannot be increased under LOCC or separable channels²⁶) can be increased under non-entangling channels. Indeed, for any positive integers k and d with k < d, there exist pure states with the Schmidt rank k that can be transformed to states with the Schmidt rank d. To see this, note that there exist pure states ψ with the Schmidt rank d but with robustness R(ψ) < 1/k. As $R(ϕk+)=1/k$⁠, Theorem 1 implies that $ϕk+$ may be transformed into such pure states ψ by non-entangling operations. This fact is formalized in the following corollary.

The above result demonstrates that the Schmidt rank is not monotonic under non-entangling channels. It is natural to ask then which other entanglement measures remain monotonic under non-entangling channels and which ones do not. We now investigate other entanglement measures that can be increased under non-entangling channels. The Rényi entropy of entanglement of order α is defined in terms of the Schmidt coefficients of pure states as

for any α ∈ [0, +∞], where H_α are the Rényi entropies

These quantities are the Schur concave as functions of the Schmidt coefficients. Hence, the function E_α is an entanglement measure for pure states for any α. Note that the Schmidt rank of a pure state is in direct correspondence with the Rényi entropy of order zero. Since we have seen that E₀ can be increased under non-entangling channels, it is natural to consider whether E_α is not monotonic under non-entangling channels for other values of α as well. The Rényi entropy of order 1 reduces to the standard entropy of entanglement

which is known to be monotonic under non-entangling channels since it coincides with the relative entropy of entanglement E_R for pure states.²³ Furthermore, note from (4) that the Rényi entropy of entanglement of order 1/2 is a monotonic function of the robustness since

so E_1/2 is also monotonic under non-entangling channels by monotonicity of R.

In the following, we show that E_α can, in fact, be increased under non-entangling channels for any α ∈ [0, 1/2), while it cannot be increased under non-entangling channels for any α ≥ 1/2. The proof of the former is actually a simple corollary of Theorem 1. To show this, it suffices to find a pure state ψ with $E1/2(ψ)=E1/2(ϕ2+)$ but that $Eα(ψ)>Eα(ϕ2+)$ for α < 1/2. Since $E1/2(ϕ2+)=1$⁠, Theorem 1 indicates that the transformation $ϕ2+→ψ$ is possible under non-entangling channels as long as E_1/2(ψ) ≤ 1.

To prove that E_α is monotonic under non-entangling channels for all α ≥ 1/2, it suffices to show that the value of E_α coincides with that of another entanglement measure that is known to be monotonic under non-entangling channels. For every α ∈ [0, +∞) and density operators ρ and σ, the Rényi α-relative entropy of ρ with respect to σ is defined as

where supp denotes the support of an operator. Note that S₁ = S is the standard relative entropy. The Rényiα-relative entropy of entanglement of a state ρ is

where the minimization is taken over all separable states σ. The Rényi α-relative entropies are known to be contractive under CPTP maps for all α ∈ [0, 2], i.e.,

for all CPTP maps Λ.²⁷ It readily follows that E_R,α is monotonic under non-entangling channels for α ∈ [0, 2]. Further properties of E_R,α have been studied in Ref. 28. In the case α = 1, it is known that E_R,1 reduces to the well-known relative entropy of entanglement E_R. Remarkably, for all pure states ψ, one has that E_R,α(ψ) = E_1/α(ψ) for all α ∈ [0, 2] (see Corollary 2 in Ref. 28).

The result of Theorem 4 has been recently proved in Ref. 28 by relating the resource theories of coherence and entanglement. However, we have independently obtained an alternative proof of this fact that uses similar methods to those in Ref. 23, where it is proven that E(ψ) = E_R(ψ). We include our Proof of Theorem 4 in  Appendix A for the interested reader.

While majorization provides necessary and sufficient conditions for conversions of pure states under LOCC, whether one can find similar necessary and sufficient conditions for convertibility of arbitrary pure states under non-entangling channels remains unknown. In the following, we provide a sufficient condition for convertibility under non-entangling channels of arbitrary pure states that is independent of the majorization criterion. We first provide the following Lemma (which was first proved in Ref. 29).

We now explore what transformations are possible under the more restrictive class of dually non-entangling channels. Recall that these are the channels Λ such that both it and its dual Λ* do not generate entanglement (although Λ* is not necessarily a channel). In particular, we show that the Schmidt rank can be increased by this more restrictive class of transformations.

We first recall some conditions for certain operators to be separable. For a pure state ψ with Schmidt coefficients λ = (λ₁, …, λ_d) (in decreasing order), it holds that $R(ψ‖1d21d2)=d2λ1λ2$ (see Appendix B of Ref. 25). Hence, operators of the form $ψ+s1d2$ for s ≥ 0 are separable if and only if $s≥λ1λ2$⁠. Additionally, for any Hermitian operator Δ on a bipartite space with local dimensions d₁ and d₂, the operator $1d1d2+Δ$ is always separable whenever ‖Δ∥ ≤ 1, where ∥·∥ is the Frobenius norm (see Theorem 1 of Ref. 30). We now state a useful condition for certain types of channels to be dually non-entangling.

Now that we have conditions for a channel of the form in (7) to be dually non-entangling, we may construct such dually non-entangling channels that increase the Schmidt rank of pure states arbitrarily. The trick is to construct pure states ψ and ϕ with Schmidt coefficients λ and μ that satisfy both (8) and (9). This is done in the Proof of Theorem 9.

There is nothing remarkable about the statement in Theorem 9 in the case when k ≥ d. However, if one chooses 2 ≤ k < d, Theorem 9 indicates that it is possible to increase the Schmidt rank of a pure state from k to d by a non-entangling channel. We remark that it does not seem possible to strengthen the above results by considering maps whose outputs are mixtures of ϕ and some separable state σ other than the identity, as we did in Theorems 1 and 7 for the case of non-entangling channels. The reason is that σ can be rank-deficient³⁰ so that there exists some other separable state $ρ∈S$ such that Tr(σρ) = 0. With this state, the output of the dual map Λ*(ρ) is entangled and thus, Λ cannot be dually non-entangling.

Note that a completely positive map Λ is separable if and only if its dual map Λ* is separable. As we have seen here, the same is not necessarily true of separability preserving maps, a fact that lead us to explore the class of maps that are dually non-entangling. Similarly, a map Λ is PPT if and only if its dual map is PPT. Indeed, for any map Λ, it is clear that $(Λ*)Γ=(ΛΓ)*$⁠. Since a map is completely positive if and only if its dual is, it follows that Λ^Γ is completely positive if and only if $(Λ*)Γ$ is. The condition that a map is PPT-preserving, however, is not equivalent to the condition that its dual map be PPT-preserving. The class of “dually PPT-preserving” maps may also be studied, in principle, but are not investigated in this paper.

We now investigate the difference between the class of PPT maps (whose Choi representation is PPT) defined by Rains¹⁴ and the class of PPT-preserving maps defined in this paper. Despite some confusion in the literature,^2,31 these classes are not equivalent⁴⁹ (this distinction goes away, however, if we require that a map be PPT-preserving when tensored with the identity on any ancilla space). We have seen that these two classes of maps are, indeed, distinct as, for example, the swap operation is clearly PPT-preserving but its Choi representation is not PPT. To further distinguish between these classes of maps, we present in this section the examples of transformations under PPT-preserving channels that are not possible under PPT channels. This will be done by investigating the negativity, a quantity defined as

for bipartite states ρ. It is well-known that the negativity is monotonic under PPT channels (and thus, under LOCC as well), so the negativity is an entanglement measure. Here, however, we show that the negativity can be increased by PPT-preserving channels. This indicates that the class of PPT-preserving channels is far more powerful for entanglement manipulation than the class of channels whose Choi representations are PPT.

Before proceeding, we first note that states of the form $a1d2/d2+(1−a)ϕd+$ are PPT if and only if a ≥ d/(d + 1). Indeed, note that

which is positive exactly when a ≥ d/(d + 1) since the eigenvalues of F_d are ±1. We now show that the negativity can be increased by PPT-preserving channels.

Note that for ρ and Λ as in the proof above, we have N(Λ(ρ))/N(ρ) = (d − 1)/2. Therefore, this ratio can be made arbitrarily big by taking a large enough d. Moreover, a PPT-preserving channel that is not PPT can be constructed from any NPT state ρ and a corresponding pure state vector |ψ⟩ corresponding to a negative eigenvalue of ρ^Γ. Indeed, for such ρ and |ψ⟩, define the operator

The corresponding channel Λ of the form in (15) is clearly PPT-preserving since it maps every state to a PPT state, which results from the fact that Tr(Aσ) ≥ d/(d + 1) for every PPT state σ. Nonetheless, this channel is not PPT since the operator J(Λ)^Γ is not a positive semi-definite. Indeed, note that $(1d2−Fd)⊗ρT≥0$ but that

since Tr(Aρ^Γ) < d/(d + 1).

In order to explore the possible existence of NPT bound entanglement under LOCC, we address the question of distillable entanglement under classes of channels that are strictly larger than LOCC. As mentioned in the Introduction, it has been shown that all NPT states are distillable by PPT channels,¹³ but non-entangling channels are independent from this class of maps. Reference 9 has shown that all entangled states are distillable not only under asymptotically non-entangling channels but also under non-entangling channels.

A state ρ is said to be distillable under some class of channels if there exists a sequence of channels {Λ_n} in this class such that $limn→∞∥Λn(ρ⊗n)−ϕ2+∥1=0$⁠, where ∥·∥₁ denotes the trace norm. It has been shown that every entangled two-qubit state is distillable by LOCC channels.³² It is tempting to argue that if some class of operations containing LOCC can convert ρ to a two-qubit entangled state, then ρ is distillable using these operations. However, this argument only holds if the class of operations is closed under tensor products. More precisely, one requires that Λ^⊗n belongs to the class whenever Λ does. Indeed, if Λ is in a class of operations that is closed under tensor products and it is the case that Λ(ρ) is a two-qubit entangled state, then one can use this class of operations to generate the n-copy state $Λ(ρ)⊗n=Λ⊗n(ρ⊗n)$ for any n. For large enough n, this can further be converted to $ϕ2+$ with arbitrary precision using the LOCC distillation protocol of Ref. 32. It is true that Λ^⊗n is LOCC or PPT whenever Λ is LOCC or PPT. However, for other classes of mappings, this will not be true (see below). We must, therefore, distinguish between the ability of a class of operations to transform ρ into an LOCC-distillable state on the single-copy level vs its ability to distill ρ into a maximally entangled state in the asymptotic limit.

Although obtaining an LOCC-distillable state is weaker than obtaining the desired state $ϕ2+$⁠, we believe that studying this question is operationally justified. LOCC operations are generally regarded as being less resource intensive since they do not require the transmission of quantum information. Thus, our approach can be seen as partitioning the distillation of $ϕ2+$ into two steps: the LOCC part and the non-LOCC part. In this way, we can minimize the quantum communication resources needed to perform the distillation by completing as much of the task as possible using LOCC. An analogous rationale is applied in other quantum resource theories such as magic-state quantum computation, in which a quantum computation is divided into two parts: the generation of the so-called “magic states” and then the manipulation of these states using much simpler stabilizer operations.³³ 

It is easy to see that not all non-entangling maps are completely non-entangling. A simple example is given by the flip map Λ(⋅) = F_d ⋅ F_d (see also Sec. VI). However, this does not forbid in principle that Λ^⊗n belongs to the class of non-entangling maps whenever Λ does. We start this section by answering this question providing an explicit example of a non-entangling map that becomes entangling when taken in a two-copy form.

Given the above result, we now turn to the problem of obtaining LOCC-distillable states on the single-copy level. Our techniques will involve maps whose outputs are always mixtures of some operator with the maximally entangled state of two qubits. Note that the state $p14/4+(1−p)ϕ2+$ is separable if and only if p ≥ 2/3.³⁷ We first show how any entangled state can be transformed to an LOCC-distillable state using dually non-entangling channels.

Finally, while it is known that all non-PPT entangled states are distillable under PPT channels, we next show that all such states can be converted to a state that is distillable under LOCC by only using channels in the more restrictive class of maps that are both PPT and dually non-entangling.

Finally, we show that all states (even separable ones) can be converted to an LOCC-distillable state under any class of entangling undetected channels.