Quantifying the unextendibility of entanglement

Entanglement is a striking feature of quantum mechanics, and it has a key property called unextendibility. In this paper, we present a framework for quantifying and investigating the unextendibility of general bipartite quantum states. First, we define the unextendible entanglement, a family of entanglement measures based on the concept of a state-dependent set of free states. The intuition behind these measures is that the more entangled a bipartite state is, the less entangled each of its individual systems is with a third party. Second, we demonstrate that the unextendible entanglement is an entanglement monotone under two-extendible quantum operations, including local operations and one-way classical communication as a special case. Normalization and faithfulness are two other desirable properties of unextendible entanglement, which we establish here. We further show that the unextendible entanglement provides efficiently computable benchmarks for the rate of exact entanglement or secret key distillation, as well as the overhead of probabilistic entanglement or secret key distillation.


I. INTRODUCTION
Quantum entanglement is a striking phenomenon of quantum mechanics and a key ingredient in many quantum information processing tasks [HHHH09,Hay16,Wil17]. Unshareability or unextendibility is one of the key features of entanglement [Wer89,DPS02,DPS04].Unextendibility asserts that the more entangled a bipartite state is, the less entangled its individual systems can be with a third party.Given that classical correlations can be shared among many parties, it emphasizes a key distinction between classical and quantum correlations.The "monogamy of quantum entanglement" [Ter04], which has been described and studied in many ways [CKW00, Ter04, KW04, Yan06, OV06, AI07, LDMH + 16, GG18], is also connected to this special characteristic of unextendibility.
In this work, we present a resource theory of unextendible entanglement by taking into account a state-dependent collection of free states, which is motivated by the fundamental importance of unextendibility in understanding quantum entanglement.In particular, our contributions are as follows: 1. We first introduce the unextendible entanglement, which is a class of entanglement measures that minimizes the generalized divergence between a given state ρ AB and any possible reduced state ρ AB ′ of an extension ρ ABB ′ of ρ AB (Section III).This method is based on the idea that a bipartite state's separate systems are less entangled with a third party, the more entangled the state is.
2. Second, we show that our measures of unextendible entanglement are monotone under twoextendible operations.This also means that they are monotone under local operations and one-way classical communication (1-LOCC) (Section III B).We also prove faithfulness and normalization of unextendible entanglement, two more desirable characteristics of the family of entanglement measures.We additionally take into account a variety of important Rényi relative entropy-based unextendible entanglement examples (Section III C), some of which are efficiently computable (Section IV).
3. Third, we show how the unextendible entanglement can be utilized to effectively establish fundamental limitations on both the overhead and rate of secret key and entanglement distillation (Section V).These most recent findings suggest that the unextendible entanglement provides a useful lens into the virtues of entanglement as a resource.
Remark 1 (Comparison of our results with [KDWW19,KDWW21]) Before we delve into the rest of the paper, let us briefly remark on the similarities and differences between our present paper and [KDWW19,KDWW21].The main aspect in which these works are similar is that they both establish ways of quantifying entanglement in terms of the notion of unextendibility.However, beyond this conceptual link, the works are completely different in their approaches and the resulting applications.
The papers [KDWW19,KDWW21] establish measures of unextendibility in terms of a comparison between the state of interest and the set of k-extendible states, with the measure being called the relative entropy of unextendibility.The consequences of this choice are that the relative entropy of unextendibility does not give bounds on the asymptotic one-way distillable entanglement or the asymptotic quantum capacity, mainly due to [KDWW21, Lemma 10], which states that the relative entropy of k-unextendibility can never be larger than log 2 k, regardless of the system size.Thus, this quantity is only applicable for providing upper bounds on the rates of these tasks in the non-asymptotic regime.
In contrast, as stated above, here we compare the bipartite state ρ AB of interest, on systems AB, to another bipartite state σ AB ′ on systems AB ′ , such that there exists a tripartite state on ABB ′ with marginals given by ρ AB and σ AB ′ .Thus, this measure, called unextendible entanglement here, is completely different from the relative entropy of unextendibility from [KDWW19,KDWW21].Furthermore, as we show in our paper, the unextendible entanglement is applicable in the asymptotic regime, giving upper bounds on the rate of probabilistic distillation of entanglement and secret key.
Thus, the approach here and the approach taken in [KDWW19,KDWW21] are complementary approaches for quantifying the unextendibility of entanglement.

II. PRELIMINARIES
We begin in this section by establishing some notation and reviewing some definitions.

A. Notations
Throughout, quantum systems are denoted by A, B, and C, have finite dimensions, and are associated with their underlying Hilbert spaces.Systems described by the same letter are assumed to be isomorphic: A 1 ∼ = A 2 and A ∼ = A ′ .The set of linear operators acting on system A is denoted by L(A), the set of Hermitian operators acting on system A is denoted by Herm(A), the set of positive semi-definite operators by P(A), and the set of quantum states (density operators) by S(A).For a linear operator L, we use L T to denote its transpose and L † to denote its conjugate transpose.The α-norm of L is defined as AB is also known as an ebit.A purification of a quantum state ρ A ∈ S(A) is a pure state Ψ ρ AR ∈ S(AR) such that Tr R Ψ ρ AR = ρ A , where R is called the purifying system.An extension of a quantum state ρ A ∈ S(A) is a quantum state (not necessarily pure) ρ AE ∈ S(AE) such that Tr E ρ AE = ρ A , where E is called the extension system.Quantum channels are completely positive (CP) and trace preserving (TP) maps from L(A) to L(B) and denoted by N A→B ∈ Q(A → B).The identity channel from L(A) to L(A) is denoted by id A .

B. Quantum divergences
Let R denote the field of real numbers.A functional D : S(A) × S(A) → R ∪ {+∞} is a generalized divergence [PV10] if for arbitrary Hilbert spaces H A and H B , arbitrary states ρ A , σ A ∈ S(A), and an arbitrary channel N A→B ∈ Q(A → B), the following data-processing inequality holds (1) The data-processing inequality above implies that there is a constant c ∈ R such that D(ρ A σ A ) ≥ c for all states ρ and σ.Indeed, we can choose the channel N A→B in (1) to be the trace and replace channel ρ → Tr[ρ]ω, where ω is a state.Then applying (1), we conclude that D(ρ A σ A ) ≥ D(ω ω).Furthermore, considering the trace and replace channel ρ → Tr[ρ]τ , where τ is a state, and applying the data-processing inequality twice in opposite directions implies that D(ω ω) = D(τ τ ) for all states ω and τ .Thus, we can set c = D(ω ω), justifying the claim, and we see that D(ρ A σ A ) takes its minimal value when the two states are equal, i.e., ρ = σ.Without loss of generality, we can then redefine the generalized divergence to be D(ρ A σ A ) − c, so that the redefined quantity satisfies D(ω ω) = 0 for every state ω.We make this assumption in what follows.
The following faithfulness property ensures that the above divergences are non-negative for normalized states and vanishes only if both arguments are equal (proof in Appendix A).It is useful for later analysis.

III. QUANTIFYING THE UNEXTENDIBILITY OF ENTANGLEMENT
This section is structured as follows.In Section III A, we define the set of free states determined by a given bipartite quantum state ρ AB .In Section III B, we introduce the generalized unextendible entanglement.In Section III C, we consider special cases of the generalized unextendible entanglement, which include those based on the quantum relative entropy [Ume62], the Petz-Rényi relative entropy [Pet86], the sandwiched Rényi relative entropy [MLDS + 13, WWY14], and the geometric Rényi relative entropy [Mat15] (see also [FF21]).We establish several desirable properties for these measures, including the fact that they are monotone under two-extendible operations.

A. State-dependent set of free states
Unlike the usual framework of quantum resource theories [CG19] and that which was established for unextendibility in [KDWW19], the free states in our resource theory are state-dependent, which is inspired by the definition and meaning of unextendibility of entanglement.Note that state-dependent resource theories were previously considered in a different context [RNBG20].To be specific, given a bipartite state ρ AB , the free states are those bipartite states that are possibly shareable between Alice and a third party B ′ , where the system B ′ is isomorphic to B. Mathematically, the set of free states corresponding to a state ρ AB is defined as follows: Furthermore, if ρ AB ∈ F ρ AB , then ρ AB is two-extendible, indicating that there exists an extension Separable states are always two-extendible.Interestingly, there are entangled states that are also two-extendible; a concrete example can be found in Eq. ( 7) of [Doh14].As so, the resource theory under consideration differs from commonly studied entanglement quantifiers where separable states are usually regarded as constituting the full set of free states.We introduce selective two-extendible operations as the free operations under consideration, which generalize the two-extendible channels previously defined and investigated in [KDWW19].Mathematically, a selective two-extendible operation consists of a set of CP maps of the following form: such that y∈Y E y AB→A ′ B ′ is trace-preserving, each map E y AB→A ′ B ′ is completely positive and twoextendible, in the sense that there exists an extension map 1. Channel extension: where 2. Permutation covariance: where W B 1 B 2 is the unitary swap channel.
The two conditions above ensure that the two marginal operations where ρ AB 1 = Tr B 2 ρ AB 1 B 2 is the marginal quantum state, are operations that are in fact equal to the original operation: We emphasize that the channel extension defined in (14) must have well defined channel marginals, which is a nontrivial property and commonly referred to as the quantum channel marginal problem.We refer interested readers to [HLA22] for comprehensive research on this problem.A two-extendible channel [KDWW19] is a special case of a selective two-extendible operation for which |Y| = 1.Now we introduce selective 1-LOCC operations as a special case of selective two-extendible operations.Mathematically, a selective 1-LOCC operation consists of a set of CP maps of the following form: where X and Y are the alphabets, each map F x,y A→A ′ is completely positive, the sum map x∈X ,y∈Y F x,y A→A ′ is trace-preserving, and each map G x,y B→B ′ is completely positive and tracepreserving.A 1-LOCC channel is a special case of a selective 1-LOCC operation for which |Y| = 1.The fact that selective 1-LOCC operations are a special case of selective two-extendible operations can be verified by constructing the extension map of L y AB→A ′ B ′ in (19) in the following way: Such a choice satisfies the channel extension and permutation covariance properties required of selective two-extendible operations.We emphasize that similar inclusions were observed in [KDWW19] for two-extendible channels and 1-LOCC channels.

B. Generalized unextendible entanglement
We introduce a family of entanglement measures called unextendible entanglement.Given a bipartite state ρ AB , the key idea behind these measures is to minimize the divergence between ρ AB and any possible reduced state ρ AB ′ of an extension ρ ABB ′ of ρ AB .These measures are intuitively motivated by the fact that the more that a bipartite state is entangled, the less that each of its individual systems is entangled with a third party.
Definition 1 (Generalized unextendible entanglement) The generalized unextendible entanglement of a bipartite state ρ AB is defined as where the infimum ranges over all free states in F ρ AB , as defined in (12). Remark , where Ψ ABC is a purification of ρ AB .Using this correspondence, the generalized unextendible entanglement can be defined in a dynamical way as where the infimum ranges over every quantum channel R C→B ′ .
In the following, we show that the generalized unextendible entanglement possesses the desirable monotonicity property required for a valid entanglement measure [HHHH09].Notice that we have only proven the monotonicity for (selective) two-extendible operations, which include (selective) 1-LOCC operations as special cases.We do not expect the unextendible entanglement to be monotone under all LOCC operations.
Theorem 2 (Two-extendible monotonicity) The generalized unextendible entanglement does not increase under the action of a two-extendible channel.That is, the following inequality holds for an arbitrary bipartite quantum state ρ AB and an arbitrary two-extendible channel E AB→A ′ B ′ : Proof.Let ρ AB 1 B 2 be an arbitrary extension of ρ AB , with ρ AB = ρ AB 1 .Since the channel E AB→AB ′ is two-extendible, there exists an extending channel 2 that satisfies the conditions stated in ( 14)-(15).In particular, the marginal channels The first inequality follows from monotonicity of the generalized channel divergence under the action of the quantum channel E AB→AB ′ .The equality follows from the observations stated above.The final inequality follows from the definition of the generalized unextendible entanglement, by applying the fact that . Since the inequality above holds for an arbitrary extension ρ AB 1 B 2 of ρ AB , we can take an infimum over all such extensions and conclude (23).
It turns out that if the underlying generalized divergence is faithful and continuous, then the generalized unextendible entanglement E u is faithful, in the sense that E u (ρ AB ) = 0 if and only if ρ AB is two-extendible.
Proposition 3 (Faithfulness) If the underlying generalized divergence is faithful, then the generalized unextendible entanglement E u is non-negative and Proof.Non-negativity of E u follows from the assumption of faithfulness.Now suppose that ρ AB is two-extendible.Then there exists an extension ρ ABB ′ such that ρ AB = ρ AB ′ .Then it follows that E u (ρ AB ) = 0 by definition and from the assumption that the underlying divergence is faithful.Now suppose that E u (ρ AB ) = 0.By the assumption of continuity, this means that there exists an extension ρ ABB ′ of ρ AB such that ρ AB = ρ AB ′ .Then, by definition, ρ AB is two-extendible.

C. α-unextendible entanglement
The α-unextendible entanglement of a bipartite state ρ AB is defined by taking the generalized divergence in (21) to be the Petz-Rényi relative entropy [Pet86], the sandwiched Rényi relative entropy D α (ω τ ) [MLDS + 13, WWY14], or the geometric Rényi relative entropy [Mat15].It is meaningful to consider the families of α-unextendible entanglement measures with different quantum divergences for several reasons.First of all, we are able prove their general properties once and for all, without analyzing them case by case.Second, some special cases of these measures lead to novel applications, as we will show in Section V.It is highly possible that more of them find operational interpretations in other tasks, using the insights learned from the proved properties.
the α-sandwiched unextendible entanglement is defined for α ∈ (0, ∞] as and the α-geometric unextendible entanglement is defined for α ∈ (0, ∞) as In the following, we show that the α-unextendible entanglement possesses the desirable monotonicity property required for an entanglement measure [HHHH09].More specifically, it is an immediate consequence of Theorem 2 and data processing that α-unextendible entanglement is monotone under the action of a two-extendible channel for certain values of α.
Notice that we have only asserted the monotonicity for two-extendible operations, including 1-LOCC operations as special cases.As already mentioned, we do not expect the unextendible entanglement to be monotone under all LOCC operations.
For certain values of α ≥ 1, the above monotonicity statement can be strengthened, in the sense that the α-unextendible entanglement is monotone under selective two-extendible operations.This property is stronger than what we previously proved in Theorem 2. As before, this implies that α-unextendible entanglement is monotone under selective 1-LOCC operations because these are a particular case of selective two-extendible operations.The proof of Theorem 5 can be found in Appendix B.
Theorem 5 (Selective two-extendible monotonicity) Let ρ AB be a bipartite state, and let E y AB→A ′ B ′ y be a selective two-extendible operation.Then the following inequality holds for the α-Petz unextendible entanglement for all α ∈ [1, 2]: where The following inequality holds for the α-sandwiched unextendible entanglement for all α ∈ [1, ∞]: The following inequality holds for the α-geometric unextendible entanglement for all α ∈ [1, 2]: Since the Petz-, sandwiched, and geometric Rényi relative entropies (and their limits in α → 1) satisfy all of the requirements in Lemma 1, we can invoke Proposition 3 and conclude the faithfulness property for all α-unextendible entanglement measures.
Before stating the next proposition, recall that the Petz-Rényi mutual information of a bipartite state ρ AB is defined as [GW15, Eq. (6.3) and Corollary 8] and the sandwiched Rényi mutual information of a bipartite state ρ AB is defined as [Bei13, GW15] where the minimization in both cases is with respect to every quantum state σ B .We also define the geometric Rényi mutual information of ρ AB in a similar way: Also, recall that the Rényi entropy of a quantum state ρ A is defined for α ∈ (0, 1) ∪ (1, ∞) as and it is defined for α ∈ {0, 1, ∞} in the limit, so that where rank(ρ A ) is the rank of ρ A .
The following lemmas relate the α-mutual information to Rényi entropy when the bipartite state is pure.The first was established in the proof of [SBW15, Proposition 13].
The following proposition uses the lemmas above to conclude that the α-unextendible entanglement reduces to Rényi entropy of entanglement for pure states.
Proposition 10 (Reduction for pure states) Let ψ AB be a pure bipartite state.Then the α-Petz unextendible entanglement reduces to the γ-Rényi entropy of entanglement: where γ = [2 − α]/α.The α-sandwiched unextendible entanglement reduces to the β-Rényi entropy of entanglement: where β = [2α−1] −1 .The α-geometric unextendible entanglement reduces to the zero-Rényi entropy of entanglement: Proof.For a pure state ψ AB , an arbitrary extension of it has the form where σ B 2 is a state of system B 2 .As such, it follows that The first equality follows from applying the definition of E α (ψ AB ), the second equality follows from the definition in (35), and the final equality follows from Lemma 7. The conclusions about E α (ψ AB ) and E α (ψ AB ) follow the same line of reasoning but using Lemma 8 and Lemma 9 instead, respectively.Following Proposition 10 and the fact that the Rényi entropy of the maximally mixed state (reduced state of Φ d AB ) is equal to log d for all values of β, we conclude the normalization property and state it formally as follows.
Proposition 11 (Normalization) Let Φ d AB be a maximally entangled state of Schmidt rank d.

D. Relative-entropy-induced unextendible entanglement
As mentioned above, the quantum relative entropy is a particular instance of the Petz-and sandwiched Rényi relative entropy, recovered by taking the limit α → 1.The α-unextendible entanglement in terms of these measures has already been defined and investigated above.However, it is notable enough that we define the quantum relative entropy induced unextendible entanglement explicitly here.
Definition 3 (Relative-entropy-induced unextendible entanglement) For a bipartite state ρ AB , the quantum relative entropy induced unextendible entanglement is defined as where D is the quantum relative entropy defined in (9).
As proved above, E u obeys the following properties: selective two-extendible monotonicity, faithfulness, reduction to the entropy of entanglement for pure bipartite states, normalization, convexity, and subadditivity.It is also efficiently computable by means of semi-definite programming, using the approach of [FSP19,FF18].
The third property mentioned above follows directly from Proposition 10 for α = 1, and it asserts that for pure bipartite states, the relative-entropy-induced unextendible entanglement evaluates to the von Neumann entropy of the reduced state.Although the proof has already been given, we can see it with a straightforward proof consisting of a few steps.Let ψ AB ≡ |ψ ψ| AB be a pure state.An arbitrary extension of ψ AB is of the form σ ABB where the second equality follows from the definition of quantum mutual information.

IV. EFFICIENTLY COMPUTABLE ENTANGLEMENT MEASURES A. Max-unextendible entanglement
Another interesting instance of the sandwiched Rényi relative entropy is the max-relative entropy, as recalled in (11).The max-relative entropy was originally defined and studied in [Dat09].
Here we adopt the max-relative entropy to define the max-unextendible entanglement.It turns out that this measure is additive and can be calculated efficiently by utilizing a semidefinite program.
Definition 4 (Max-unextendible entanglement) For a given bipartite state ρ AB , the maxunextendible entanglement is defined as where D max is the max-relative entropy defined in (11).
Note that the infimum in (58) can be replaced with a minimum.
From the definition of max-unextendible entanglement, it follows that it can be computed efficiently by means of a semidefinite program (SDP).To be more specific, the following two optimization programs satisfy strong duality and both evaluate to 2 −2E u max (ρ AB ) .
Primal Program The primal SDP follows by considering that The dual SDP can be obtained by standard methods (e.g., the Lagrange multiplier method).For completeness, we provide a proof in Appendix F. By using the primal and dual expressions of 2 −2E u max (ρ AB ) and strong duality, it follows that E u max (ρ AB ) is additive (proof in Appendix G), which is an appealing feature that finds use in Section V.

Proposition 14 (Additivity) Let
where the entanglement is evaluated across the A:B bipartition.

B. Min-unextendible entanglement
In this section, we consider the limit of the Petz-Rényi relative entropy as α → 0, which is known as the min-relative entropy [Dat09, Definition 2].Let us first recall the definition.Let ρ ∈ S(A) and σ ∈ P(A).Let Π ρ denote the projection onto the support of ρ.Then the min-relative entropy of ρ and σ is defined as With D min , we define the min-unextendible entanglement as follows.
Definition 5 (Min-unextendible entanglement) For a given bipartite state ρ AB , the minunextendible entanglement is defined as Note that the infimum in (64) can be replaced with a minimum.Much like the max-unextendible entanglement, the min-unextendible entanglement can also be calculated as the solution to a semidefinite program.The following two optimization programs satisfy strong duality, and both evaluate to 2 −2E u min (ρ AB ) .We derive the dual SDP in Appendix F.
Following similarly to the proof of Proposition 14, we can show that the min-unextendible entanglement is additive.
where the entanglement is evaluated across the A:B bipartition.
Furthermore, the min-unextendible entanglement of a pure bipartite state can be computed explicitly (proof in Appendix H).

Proposition 16 Let |ψ
Then the following equality holds Interestingly, E u min (ψ AB ) has an operational interpretation in terms of deterministic entanglement transformation [DFJY05], which we briefly introduce as follows.Let |ψ 1 and |ψ 2 be two pure bipartite states for systems AB.Let m ∈ N be an integer.We define f (m) to be the maximum integer n such that ψ ⊗m 1 can be transformed into ψ ⊗n 2 by LOCC deterministically.The deterministic entanglement transformation rate from ψ 1 to ψ 2 , written D(ψ 1 → ψ 2 ), is defined as Intuitively, for sufficiently large m, one can transform m copies of ψ 1 exactly into mD(ψ 1 → ψ 2 ) copies of ψ 2 by LOCC.We have the following proposition, which is a consequence of Proposition 16 and the developments in [DFJY05]: Proposition 17 Let |ψ AB be a pure state in AB and Φ 2 be the ebit.Then the following equality holds C. Unextendible fidelity Let ρ, σ ∈ S(A) be two quantum states.The (root) fidelity between ρ and σ is defined as Here we define the unextendible fidelity of a state ρ AB : Definition 6 (Unextendible fidelity) For a given bipartite state ρ AB , the unextendible fidelity is defined as Note that the supremum in (72) can be replaced with a maximum.Suppose that Ψ ABC is a purification of ρ AB .By applying Remark 3, we see that the unextendible fidelity can be alternatively understood as a measure of how well one can recover the state ρ AB if system B is lost and a recovery channel is performed on the purification system C alone, due to the following equivalent formulation: where the maximum ranges over every quantum channel N C→B ′ .The unextendible fidelity is thus similar in spirit to the fidelity of recovery from [SW15], but one finds that it is a different measure when analyzing it in more detail.By examining (4) and (71), one immediately finds that D 1/2 (ρ σ) = − log[F (ρ, σ)] 2 .Thus, we establish the following equivalence between unextendible fidelity and 1/2-sandwiched unextendible entanglement: Since the fidelity function is SDP computable [Wat13], it follows that the unextendible fidelity can be computed efficiently by means of a semidefinite program.To be more specific, the following two optimization programs satisfy strong duality and both evaluate to F u (ρ AB ).For completeness, we show in detail how to derive the dual program in Appendix I.
We also establish the following equivalent dual representation of As a direct consequence of this equivalent dual representation, we find that the extendible fidelity is multiplicative (proof in Appendix J).
Proposition 18 (Multiplicativity) Let ρ A 1 B 1 ∈ S(A 1 B 1 ) and ρ A 2 B 2 ∈ S(A 2 B 2 ) be two bipartite states.The following equality holds where the entanglement is evaluated across the A:B bipartition.
2 ) be two bipartite states.The following additivity relation holds where the entanglement is evaluated across the A:B bipartition.
Using the relation in ( 74)-(76) between the unextendible fidelity and the 1/2-sandwiched unextendible entanglement, we find that the unextendible fidelity of a pure bipartite state can be computed explicitly.
Proposition 20 Let ψ AB be a pure bipartite state.Then the following equality holds where λ max (ρ A ) is the maximal eigenvalue of ψ A .

V. APPLICATIONS FOR SECRET KEY AND ENTANGLEMENT DISTILLATION A. Private states and unextendible entanglement
In this section, we review the definition of a private state [HHHO05, HHHO09], and then we establish a bound on the number of private bits contained in a private state, in terms of the state's unextendible entanglement.These results find applications in the next two subsections (Sections V B and V C), where we investigate the overhead of probabilistic secret key distillation and the rate of exact secret key distillation.
We first review the definition of a private state [HHHO05,HHHO09].Let ρ ABA ′ B ′ ∈ S(ABA ′ B ′ ) be a state shared between spatially separated parties Alice and Bob, where Alice possesses systems A and A ′ and Bob possesses systems B and B ′ , such that where M(•) = k |k k|(•)|k k| is a projective measurement channel and σ E is some state on the purifying system.The systems A and B are known as key systems, and A ′ and B ′ are known as shield systems.Interestingly, it was shown that a private state of log K private bits can be written in the following form [HHHO05, HHHO09] where Φ K AB is a maximally entangled state of Schmidt rank K, the state σ A ′ B ′ is an arbitrary state, and is a controlled unitary known as a "twisting unitary," with each U ij A ′ B ′ a unitary.We now establish some bounds on the number of private bits contained in a private state, in terms of its unextendible entanglement.Recall that the generalized mutual information of a state ρ AB is defined as follows [SW12]: where D is the generalized divergence discussed in Section III B and the infimum is with respect to every density operator σ B .We first show that the unextendible entanglement of a private state is bounded from below by the generalized mutual information of Φ AB (proof in Appendix K).
Proposition 21 For a γ-bipartite private state of the form in (85), the following bound holds where I D (A; B) Φ is evaluated with respect to the state Φ K AB in (85).
As a corollary of Proposition 21 and Lemmas 7, 8, and 9, we find that if the generalized divergence is set to be the Petz-, sandwiched, or geometric Rényi relative entropy, then the unextendible entanglement of a γ-bipartite private state is bounded from below by the amount of secret key that can be extracted from the state.Note that this corollary includes the relative entropy, the min-relative entropy, and the max-relative entropy as limiting cases.
Corollary 22 If the generalized divergence is the Petz-, sandwiched, or geometric Rényi relative entropy with α set so that the data-processing inequality is satisfied, then the following bound holds

B. Overhead of probabilistic secret key distillation
This section considers the overhead of probabilistic secret key distillation under selective twoextendible operations (which include 1-LOCC operations as a special case).We first formally define the overhead of probabilistic secret key distillation: Definition 7 The overhead of distilling k private bits from several independent copies of a bipartite state ρ AB is given by the minimum number of copies of ρ AB needed, on average, to produce some private state γ k ABA ′ B ′ with log K = k using 1-LOCC operations: We can also define the overhead when selective two-extendible operations are allowed: Note that it is not necessary to produce a particular private state γ k ABA ′ B ′ , but rather just some private state γ k ABA ′ B ′ having k private bits.The following inequality is a trivial consequence of definitions and the fact that a selective 1-LOCC operation is a special kind of selective two-extendible operation: It turns out that the relative-entropy-induced unextendible entanglement E u (ρ AB ), given in Definition 3, provides a lower bound on the overhead of probabilistic secret key distillation of a bipartite state ρ AB (proof in Appendix L).
Theorem 23 For a bipartite state ρ AB , the overhead of distilling k private bits from ρ AB is bounded from below by where the relative-entropy-induced unextendible entanglement is given in Definition 3.
In Theorem 5 and Proposition 13, we proved that the α-Petz unextendible entanglement for α ∈ [1, 2], the α-sandwiched unextendible entanglement for α ∈ [1, ∞], and the α-geometric unextendible entanglement for α ∈ [1, 2] satisfy selective two-extendible monotonicity and subadditivity, respectively.That is to say, all these entanglement measures can be used as lower bounds in (93).Since these divergences are monotonically increasing in α [Tom15] and D(ω τ ) ≤ D(ω τ ) [HP91], E u (ρ AB ) is the smallest unextendible entanglement measure among these choices and yields the tightest lower bound.Furthermore, the authors of [FSP19] proposed a method to accurately approximate the quantum relative entropy via semidefinite programming.This enables us to estimate the lower bound using available semidefinite programming solvers.See also [FF18] in this context.

C. Exact secret key distillation
We can also consider the setting in which the goal is to distill secret key exactly from a bipartite state by using two-extendible or 1-LOCC operations.Though exact distillation is less realistic than the above probabilistic scenario, it is still a core part of zero-error quantum information theory [GdAM16].
The one-shot 1-LOCC exact distillable key of ρ AB is defined to be the maximum number of private bits achievable via a 1-LOCC channel; that is, where γ k Â BA ′ B ′ is any private state with k private bits.The 1-LOCC exact distillable key of a state ρ AB is then defined as the regularization of K (1) 1-LOCC (ρ AB ): Note that the following inequality holds as a direct consequence of definitions: The one-shot two-extendible exact distillable key and two-extendible exact distillable key of ρ AB are defined similarly: and the following inequality holds as a direct consequence of definitions: Immediate consequences of definitions and the fact that 1-LOCC operations are contained in the set of two-extendible operations are the following inequalities: It turns out that the min-unextendible entanglement E u min (ρ AB ) serves as an upper bound on the two-extendible exact distillable key of a bipartite state ρ AB (proof in Appendix M).
Theorem 24 For a bipartite state ρ AB , its asymptotic exact distillable key under two-extendible operations is bounded from above as Actually, it is evident from the proof that in Theorem 24, both E u max (ρ AB ) and E u 1/2 (ρ AB ) are valid upper bounds on K 2-EXT (ρ AB ), since they are also monotonic and additive.However, since the min-unextendible entanglement E u min (ρ AB ) leads to the tightest upper bound.Note that the inequalities in (102) follow from the facts that the fidelity obeys data processing under the channel

D. Overhead of probabilistic entanglement distillation
Entanglement distillation aims at obtaining maximally entangled states from less entangled bipartite states shared between two parties via certain free operations.As a central task in quantum information processing, various approaches [VP98, Rai99, VW02, Rai01, HHH00, CW04, WD17b, LDS17, WD16, FWTD19, KDWW19, RFWG19] have been developed to characterize and approximate the performance of the rates of deterministic entanglement distillation.
Here, we consider the overhead of probabilistic entanglement distillation [BDSW96, CCGFZ99, PSBZ01, NFB14, CB08, RST + 18] under selective two-extendible operations, similar to what we considered for probabilistic secret key distillation.We begin by defining the overhead of probabilistic entanglement distillation.
Definition 8 The overhead of distilling m ebits from several independent copies of a bipartite state ρ AB is equal to the minimum number of copies of ρ AB needed, on average, to produce m copies of the ebit Φ 2 using selective 1-LOCC operations: We can also define the overhead when selective two-extendible operations are allowed: The following inequality is a trivial consequence of definitions and the fact that a selective 1-LOCC operation is a special kind of selective two-extendible operation: Since secret key can be obtained from ebits [HHHO05] via local operations, a direct corollary of Theorem 23 is that E u is a lower bound on the overhead of probabilistic entanglement distillation.
Corollary 25 For a bipartite state ρ AB , the overhead of distilling m ebits from ρ AB is bounded from below as where E u (ρ AB ) is the relative-entropy-induced unextendible entanglement from Definition 3.

E. Exact entanglement distillation
The one-shot 1-LOCC exact distillable entanglement of ρ AB is defined to be the maximum number of ebits achievable via a 1-LOCC channel; that is, where Φ d Â B is a maximally entangled state of Schmidt rank d.The 1-LOCC exact distillable entanglement of a state ρ AB is then defined as the regularization of E (1) 1-LOCC (ρ AB ): Note that the following inequality is a direct consequence of the definitions: The one-shot two-extendible exact distillable entanglement and two-extendible exact distillable entanglement of ρ AB are defined similarly: and the following inequality is a direct consequence of definitions: Immediate consequences of definitions and the fact that 1-LOCC operations are contained in the set of two-extendible operations are the following inequalities: As a direct corollary of Theorem 24 and the relation between private states and ebits mentioned previously (just below (105)), the min-unextendible entanglement E u min serves as an upper bound on the exact distillable entanglement via two-extendible operations.
Corollary 26 For a bipartite state ρ AB , its asymptotic exact distillable entanglement under twoextendible operations is bounded from above by its min-unextendible entanglement:

F. Examples
In this section, we apply our bounds on the overhead of probabilistic entanglement or secret key distillation to three classes of states: isotropic states, S states, and erased states.We compare our lower bounds and other known estimations of the overhead to upper bounds derived from known distillation protocols.In particular, we show that our lower bound on the overhead of distillation is tight for erased states.
To begin with, recall the relative entropy of entanglement [VP98]: where SEP denotes the set of separable states.It is known that the relative entropy of entanglement is monotone under selective LOCC [VP98].Thus, the relative entropy of entanglement E R (ρ AB ) can be used to estimate the overhead of distillation under selective LOCC via an approach similar to that given in the proof of Corollary 25.
Isotropic states: Let us first investigate the class of isotropic states ρ r , defined as [HH99] where r ∈ [0, 1], so that ρ r is a convex mixture of a maximally entangled state Φ d of Schmidt rank d and its orthogonal complement.Numerous works have been carried out to study the Here we investigate probabilistic entanglement distillation under 1-LOCC operations and show the advantage of our method in estimating the overhead.We consider the case d = 2 for simplicity.Figure 1 plots the unextendible entanglement and relative entropy of entanglement for the overhead of probabilistic entanglement or secret key distillation for this set of states.1Note that the relative entropy of entanglement for qubit-qubit isotropic states (d = 2) was calculated in [VPRK97].S states: An S state is a mixture of the Bell state and non-orthogonal product noise, whose distillation protocols were studied in [RST + 18, ZZW + 21].Here, we define it as Figure 2 plots the unextendible entanglement and relative entropy of entanglement for the overhead of probabilistic entanglement or secret key distillation for this set of states.The result shows that our result establishes a better lower bound on the overhead of probabilistic distillation under 1-LOCC operations.
Erased states: We also consider the class of erased states, which are the Choi states of quantum erasure channels [GBP97]: where ε ∈ [0, 1], Φ 2 AB is an ebit, and |e is some state that is orthogonal to π A .For simplicity, we choose d A = d B = 2 (qubit system), and d A ′ = d A + 1 = 3.This state can be obtained as follows.Alice and Bob share a two-qubit maximally entangled state Φ AB and Alice transmits her local copy through an erasure channel, so that with probability 1 − ε it is unaffected and with probability ε it is "erased" (replaced with the erasure flag state e) .
Interestingly, there is a simple formula for the relative-entropy-induced unextendible entanglement of the erased state (proof in Appendix N).
Proposition 27 The unextendible entanglement of the erased state Corollary 25 together with Proposition 27 shows that 1/E u (ρ ε A ′ B ) = 1/(1 − ε) is a lower bound on the overhead of probabilistically distilling one ebit from ρ ε A ′ B .Indeed, we can show the tightness of the lower bound by considering the following one-way LOCC protocol that achieves this bound.Given ρ ε A ′ B , Alice detects whether her local system is erased or not by performing a binary measurement {1 A , |e e|}.With probability 1 − ε, she finds that the system is not erased.Then she sends this information to Bob, and the protocol finishes with one ebit shared between them.As such, we conclude the following.
Theorem 28 For the erased state ρ ε A ′ B , we have Therefore, our estimation on the overhead of distilling ebits from erased states under selective one-way LOCC is optimal in the sense that the upper bound on the overhead of the above protocol matches our lower bound from Corollary 25, thus characterizing the ability of probabilistic distillation for erased states.Note that this result can be generalized to multiple copies of the erased state.
We further consider the overhead of distillation under LOCC operations for the erased states in the following proposition (proof in Appendix O), indicating an interesting fact that the optimal protocol for distilling one ebit from erased states under one-way LOCC operations matches the lower bound 1/E R (ρ ε A ′ B ) ≡ 1/(1 − ε) on the overhead under LOCC operations.It reveals that one-way LOCC operations have the same power and performance as LOCC operations in probabilistically distilling ebits from erased states.
Proposition 29 The relative entropy of entanglement of the erased state

VI. CONCLUDING REMARKS
Our work introduces a family of entanglement measures called unextendible entanglement to quantify and investigate the unextendibility of entanglement.The crucial technical contribution was intuitively motivated by the fact that the more entangled a bipartite state is, the less each of its constituent systems can be entangled with a third party.A key distinction to previous works is that our proposed measures restrict the free set to be state dependent.These entanglement measures have desirable properties, including monotonicity under selective two-extendible operations, faithfulness, and normalization.The unextendible entanglement in our work has also found direct operational applications in evaluating the overhead and rate of entanglement or secret key distillation.As a notable application example, we characterized the optimal overhead of distilling one ebit from the erased states under one-way LOCC operations.We also note that the probabilistic state conversion task, which has already been studied for the pure-state case [Vid99, AOP16], can be further explored using the results in this paper.
An important problem for future work is to investigate to what extent the bounds on the asymptotic distillable secret key or entanglement can be approached.One potential approach is to explore the connections to distinguishability distillation (i.e., hypothesis testing) [Nag06, Hay07, ANSV08, MO15, WW19].It is also of interest to consider an extension to the resource theory of k-unextendibility and entanglement dilution [BDSW96, HHT01, VDC02, WD17a, BD11, WW23], where the techniques applied in [XFWD17] might be useful here.Moreover, we have only considered the extendibility on system B when defining generalized unextendible entanglement measure, yielding an asymmetric entanglement measure.It would be meaningful to further explore the extendibility on system A and promote our measure to a symmetric one.Another interesting direction is to develop the continuous-variable setting, where the results of [LKAW19] could be helpful, as well as the dynamical setting of quantum channels.
ρ AB 1 ≤ tρ AB 2 For every y such that p(y) > 0, due to complete positivity and two-extendibility of the map E y AB→A ′ B ′ , we have that Therefore, In the above, (B26) is due to (B23), while (B27) is due to the concavity of the logarithm.
Appendix C: Proof of Lemma 8 Without loss of generality, let σ B be an arbitrary state with support equal to the support of the reduced state ψ B .Then we have that For every pure bipartite state ψ AB , there exists an operator where {|i A } i and {|i B } i are orthonormal bases.Furthermore, by taking the polar decomposition of X A , there exists a unitary U A and a density operator ρ A (having the same spectrum as ψ A ) such that X A = U A √ ρ A .Then consider that the last line above is equal to In the above, T (σ) denotes the transpose of the state σ.Now taking a minimum over every state σ and applying [MLDS + 13, Lemma 12], we find that min The conclusion that H Taking an infimum over all extensions σ A 1 B 1 B ′ 1 and σ A 2 B 2 B ′ 2 leads to the inequality in (53).The inequalities in (54) and (55) are established similarly.

Appendix F: Derivations of dual SDPs and strong duality for max-and min-unextendible entanglement
We first derive the dual SDP in (60), by means of the Lagrange multiplier method.Consider that sup The first equality follows by introducing the Lagrange multipliers X AB ≥ 0 and Y AB ∈Herm.Indeed, the constraint λρ AB ≤ Tr B [σ ABB ′ ] does not hold if and only if inf and the constraint The second equality follows from basic algebra.The inequality follows from the max-min inequality.The third equality follows by intepreting λ, σ and 2 ) forms a dual feasible solution.Thus which gives E u max (ρ Appendix H: Proof of Proposition 16 This proposition follows from (45) by taking the limit α → 0. Alternatively, we briefly outline another proof as follows.For a pure state ψ AB , an arbitrary extension of it has the form σ Here we derive the dual SDP of the primal SDP in (77) for unextendible fidelity.We follow an argument similar to that given in [BT16].We first bring the primal program into standard form, which expresses the primal problem as a maximization of the functional Tr[AX] over X ≥ 0, subject to the constraint Φ(X) = B. Hence, we set

(I4)
The Slater condition for strong duality is satisfied, using the fact that the primal problem is feasible and the dual problem is strictly feasible.To see this, let σ B ′ be a quantum state.Then the operator is primal feasible since X ≥ 0 and Φ( X) = B.For the dual problem, the operator is strictly feasible since For a given bipartite state ρ AB , suppose that there is a two-extendible channel Λ AB→ Â B that transforms ρ AB to a private state γ k Â BA ′ B ′ with k private bits.By the monotonicity of the minunextendible entanglement, the following inequality holds where the last inequality follows from Corollary 22. Therefore, by optimizing over all two-extendible protocols, it follows that the one-shot exact distillable entanglement of ρ AB is bounded as Applying the same reasoning to the tensor-power state ρ ⊗n AB , we find that where the final equality is a consequence of Proposition 15.
Appendix N: Proof of Proposition 27 Note that the erased state ρ ε A ′ B can be written in the following direct-sum form: From this fact, one can see that each extension ρ A ′ BB ′ of ρ ε A ′ B has the form such that Tr B ′ σ ABB ′ = Φ 2 AB and Tr B ′ σ BB ′ = π B .Furthermore, any extension of Φ 2 AB necessarily has the form σ ABB ′ := Φ 2 AB ⊗ τ B ′ , where τ B ′ is a state on system B ′ .Thus where I(A; B) ρ is the quantum mutual information of ρ AB .
divergences III.Quantifying the unextendibility of entanglement A. State-dependent set of free states B. Generalized unextendible entanglement C. α-unextendible entanglement D. Relative-entropy-induced unextendible entanglement IV.Efficiently computable entanglement measures A. Max-unextendible entanglement B. Min-unextendible entanglement C. Unextendible fidelity V. Applications for secret key and entanglement distillation A. Private states and unextendible entanglement a A preliminary version of this paper has been published in the Proceedings of the 2020 IEEE International Symposium on Information Theory (ISIT 2020) [WWW20].

FIG. 2 .
FIG.2.Bounds on the overhead of distilling one ebit or one private bit from S states.For the overhead of probabilistic distillation of entanglement or secret key under 1-LOCC operations, our lower bound (solid line) outperforms the previous lower bound (dashed line) based on relative entropy of entanglement.
[CW04]nection to joinability) In [JV13, Definition 1], the concept of joinability of two bipartite states was considered.That is, two states σ AB and τ AB ′ are said to be joinable if there exists a tripartite state ω ABB ′ such thatTr B ′ [ω ABB ′ ] = σ AB and Tr B [ω ABB ′ ] = τ AB ′ .With this in mind, the measure in (21) can be understood as quantifying how unjoinable a bipartite state ρ AB is with other bipartite states of systems A and B ′ .In[CW04], it was established that a state ρ ABB ′ is an extension of ρ AB if and only if there exists a quantum channel [HHHO05,HHHO09] B ′ is called a private state[HHHO05,HHHO09]if Alice and Bob can extract a secret key from it by performing local measurements on A and B, such that the key is product with an arbitrary purifying system of ρ ABA ′ B ′ .That is, ρ ABA ′ B ′ is a private state of log K private bits if, for every purification ϕ ρ ABA ′ B ′ E of ρ ABA ′ B ′ , the following holds: Lower bounds on the overhead of distilling one ebit or one private bit from isotropic states.For the overhead of probabilistic distillation of entanglement or secret key under 1-LOCC operations, our lower bound (solid line) outperforms the previous lower bound (dashed line) based on relative entropy of entanglement.distillationrate of isotropic states under various scenarios [FWTD19, WFD18, LDS17, RST + 18].