One-shot rates for entanglement manipulation under non-entangling maps
aa r X i v : . [ qu a n t - ph ] A p r One-shot rates for entanglement manipulation undernon-entangling maps
Fernando G.S.L. Brand˜ao and Nilanjana Datta
Abstract —We obtain expressions for the optimal rates of one-shot entanglement manipulation under operations which generatea negligible amount of entanglement. As the optimal rates forentanglement distillation and dilution in this paradigm, weobtain the max- and min-relative entropies of entanglement,the two logarithmic robustnesses of entanglement, and smoothedversions thereof. This gives a new operational meaning to theseentanglement measures. Moreover, by considering the limit ofmany identical copies of the shared entangled state, we partiallyrecover the recently found reversibility of entanglement manipu-lation under the class of operations which asymptotically do notgenerate entanglement.
I. I
NTRODUCTION
In the distant laboratory paradigm of quantum informationtheory, a system shared by two or more parties might have cor-relations that cannot be described by classical shared random-ness; we say a state is entangled if it contains such intrinsicallyquantum correlations and hence cannot be created by localoperations and classical communication (LOCC). Quantumteleportation [1] shows that entanglement can actually be seenas a resource under the constraint that only LOCC operationsare accessible. Indeed, one can use entanglement and LOCCto implement any operation allowed by quantum theory [1].The development of entanglement theory is thus centered inunderstanding, in a quantitative manner, the interconversion ofone entangled state into another by LOCC, and their use forvarious information-theoretic tasks [2], [3].In [4], Bennett et al proved that entanglement manipula-tions of bipartite pure states, in the asymptotic limit of anarbitrarily large number of copies of the state, are reversible.Given two bipartite pure states | ψ AB i and | φ AB i , the formercan be converted into the latter by LOCC if, and only if, E ( | ψ AB i ) ≥ E ( | φ AB i ) , where E is the von Neumann entropyof either of the two reduced density matrices of the state.For mixed bipartite states, it turns out that the situation israther more complex. For instance there are examples of mixedbipartite states, known as bound entangled states [5], whichrequire a non-zero rate of pure state entanglement for theircreation by LOCC in the limit of many copies, but from whichno pure state entanglement can be extracted [5], [6], [7]. Fernando Brand˜ao ([email protected]) is at the Physics De-partment of Universidade Federal de Minas Gerais, Brazil. Nilanjana Datta([email protected]) is in the Statistical Laboratory, Dept. of AppliedMathematics and Theoretical Physics, University of Cambridge. This work ispart of the QIP-IRC supported by the European Community’s Seventh Frame-work Programme (FP7/2007-2013) under grant agreement number 213681,EPSRC (GR/S82176/0) as well as the Integrated Project Qubit Applications(QAP) supported by the IST directorate as Contract Number 015848’. FB issupported by a ”Conhecimento Novo” fellowship from Fundac¸˜ao de Amparoa Pesquisa do Estado de Minas Gerais (FAPEMIG).
This inherent irreversibility in the asymptotic manipulationof entanglement led to the exploration of different scenariosfor the study of entanglement, departing from the original onebased on LOCC operations (see e.g. [8], [9], [10], [11], [12],[13]). The main motivation in these studies was to develop asimplified theory of entanglement manipulation, with the hopethat it would also lead to new insights into the physicallymotivated setting of LOCC manipulations.Recently one possible such scenario has been identified.In Refs. [14], [15], [16] the manipulation of entanglementunder any operation which generates a negligible amount ofentanglement, in the limit of many copies, was put forward.Remarkably, it was found that one recovers for multipartitemixed states the reversibility encountered for bipartite purestates under LOCC. In such a setting, only one measure ismeaningful: the regularized relative entropy of entanglement[17], [18]; it completely specifies when a multipartite state canbe converted into another by the accessible operations. Thisframework has also found interesting applications to the LOCCparadigm, such as a proof that the LOCC entanglement costis strictly positive for every multipartite entangled state [16],[19] (see [20] for a different proof), new insights into sep-arability criteria [21], and impossibility results for reversibletransformations of pure multipartite states [22].In this paper we analyze entanglement conversion of generalmultipartite states under non-entangling and approximatelynon-entangling operations in the single copy regime (see e.g.[23], [24], [25], [26], [27] for other studies of the single copyregime in classical and quantum information theory). We willidentify the single copy cost and distillation functions undernon-entangling maps with the two logarithmic robustnessesof entanglement [30], [31], [28] (one of them also referredto as the max-relative entropy of entanglement [32]), and themin-relative entropy of entanglement [32], respectively. Onone hand, our findings give operational interpretation to theseentanglement measures. On the other hand, they give furtherinsight into the reversibility attained in the asymptotic regime.Indeed, we will be able to prove reversibility, under catalyticentanglement manipulations, by taking the asymptotic limit inour finite copy formulae and using a certain generalization ofquantum Stein’s Lemma proved in Ref. [19] (which is also themain technical tool used in [14], [15], [16]). We hence partiallyrecover the results of [14], [15], [16], where reversibility wasproved without the use of entanglement catalysis.The paper is organized as follows. In Section II we introducethe necessary notation and definitions. Section III contains ourmain results, stated as Theorems 1-4. These theorems are thenproved in Sections IV,V, VI and VII, respectively.
II. N
OTATION AND D EFINITIONS
Let B ( H ) denote the algebra of linear operators acting on afinite–dimensional Hilbert space H , and let B + ( H ) ⊂ B ( H ) denote the set of positive operators acting in H . Let D ( H ) ⊂B + ( H ) denote the set of states (positive operators of unittrace).Given a multipartite Hilbert space H = H ⊗ ... ⊗ H m , wesay a state σ ∈ D ( H ⊗ ... ⊗ H m ) is separable if there arelocal states σ kj ∈ D ( H k ) and a probability distribution { p j } such that σ = X j p j σ j ⊗ ... ⊗ σ mj . (1)We denote the set of separable states by S .For given orthonormal bases {| i A i} di =1 and {| i B i} di =1 inisomorphic Hilbert spaces H A and H B of dimension d , amaximally entangled state (MES) of rank M ≤ d is givenby | Ψ ABM i = 1 √ M M X i =1 | i A i| i B i . We define the fidelity of two quantum states ρ , σ as F ( ρ, σ ) = (cid:18) Tr q √ σρ √ σ (cid:19) . (2)Finally, we denote the support of an operator X by supp(X) .Throughout this paper we restrict our considerations to finite-dimensional Hilbert spaces, and we take the logarithm to base . In [35] two generalized relative entropy quantities, referredto as the min- and max- relative entropies, were introduced.These are defined as follows. Definition 1.
Let ρ ∈ D ( H ) and σ ∈ B + ( H ) be such that supp( ρ ) ⊆ supp( σ ) . Their max-relative entropy is given by D max ( ρ || σ ) := log min { λ : ρ ≤ λσ } , (3) while their min-relative entropy is given by D min ( ρ || σ ) := − log Tr (cid:0) Π ρ σ (cid:1) , (4) where Π ρ denotes the projector onto supp( ρ ) . As noted in [35], [26], D min ( ρ || σ ) is the relative R´enyientropy of order .In [32] two entanglement measures were defined in termsof the above quantities. Definition 2.
The max-relative entropy of entanglement of ρ ∈D ( H ) is given by E max ( ρ ) := min σ ∈S D max ( ρ || σ ) , (5) while its min-relative entropy of entanglement is given by E min ( ρ ) := min σ ∈S D min ( ρ || σ ) , (6) Note that D min ( ρ || σ ) is well-defined whenever supp ( ρ ) ∩ supp ( σ ) is notempty. It turns out [32] that E max ( ρ ) is not really a new quantity,but is actually equal to the logarithmic version of the globalrobustness of entanglement, given by [28] LR G ( ρ ) := log(1 + R G ( ρ )) , (7)where R G ( ρ ) is the global robustness of entanglement [30],[31] defined as R G ( ρ ) := min s ∈ R (cid:16) s ≥ ∃ ω ∈ D s . t .
11 + s ρ + s s ω ∈ S (cid:17) . (8)Another quantity of relevance in this paper is the robustnessof entanglement [30], denoted by R ( ρ ) . Its definition isanalogous to that of R G ( ρ ) except that the states ω in Eq.(8)are restricted to separable states. Its logarithmic version isdefined as follows. Definition 3.
The logarithmic robustness of entanglement of ρ ∈ D ( H ) is given by LR ( ρ ) := log(1 + R ( ρ )) . (9)We also define smoothed versions of the quantities weconsider as follows (see also [19], [36]). Definition 4.
For any ε > , the smooth max-relative entropyof entanglement of ρ ∈ D ( H ) is given by E ε max ( ρ ) := min ¯ ρ ∈ B ε ( ρ ) E max (¯ ρ ) , (10) where B ε ( ρ ) := { ¯ ρ ∈ D ( H ) : F (¯ ρ, ρ ) ≥ − ε } .The smooth logarithmic robustness of entanglement of ρ ∈D ( H ) in turn is given by LR ε ( ρ ) := min ¯ ρ ∈ B ε ( ρ ) LR (¯ ρ ) . (11) Finally, the smooth min-relative entropy of entanglement of ρ ∈ D ( H ) is defined as E ε min ( ρ ) := max ≤ A ≤ I Tr( Aρ ) ≥ − ε min σ ∈S ( − log Tr( Aσ )) . (12)We note that the definition of E ε min ( ρ ) which we use inthis paper is different from the one introduced in [32], wherethe smoothing was performed over an ε -ball around the state ρ , in analogy with the smooth version of E ε max ( ρ ) givenabove. Note also that while this new smoothing is a prioriinequivalent to the one in [32], it is equivalent to the “operator-smoothing” introduced in [25], which, in addition, gives riseto a continuous family of smoothed relative R´enyi entropies.We will consider regularized versions of the smooth min-and max-relative entropies of entanglement E ε min ( ρ ) := lim inf n →∞ n E ε min ( ρ ⊗ n ) , E ε max ( ρ ) := lim sup n →∞ n E ε max ( ρ ⊗ n ) , (13)and the quantities E min ( ρ ) := lim ε → E ε min ( ρ ) E max ( ρ ) := lim ε → E ε max ( ρ ) (14) In [19], [32] it was proved that E max ( ρ ) is equal to theregularized relative entropy of entanglement [17], [18] E ∞ R ( ρ ) := lim n →∞ n E R ( ρ ⊗ n ) , (15)where E R ( ω ) := min σ ∈S S ( ω || σ ) , (16)is the relative entropy of entanglement and S ( ω || σ ) :=Tr( ρ (log( ρ ) − log( σ ))) the quantum relative entropy.In this paper we prove that also E min ( ρ ) is equal to E ∞ R ( ρ ) (see Theorem 4).We can now be more precise about the classes of maps weconsider for the manipulation of entanglement, introduced in[14], [15]. Definition 5.
A completely positive trace-preserving (CPTP)map Λ is said to be a non-entangling (or separability preserv-ing) map if Λ( σ ) is separable for any separable state σ . Wedenote the class of such maps by SEPP . Definition 6.
For any given δ > we say a map Λ is a δ -non-entangling map if R G (Λ( σ )) ≤ δ for every separable state σ .We denote the class of such maps by δ -SEPP. In the following sections we will consider entanglement ma-nipulations under non-entangling and δ -non-entangling maps.We first give the definitions of achievable and optimal ratesof entanglement manipulation protocols under a general classof maps, in order to make the subsequent discussion moretransparent. In the definitions we will consider maps froma multipartite state to a maximally entangled state and vice-versa. It should be understood that the first two parties sharethe maximally entangled state, while the quantum state of theother parties is trivial (one-dimensional). Definition 7.
The one-shot entanglement cost of ρ under theclass of operations Θ is defined as E (1) ,εC, Θ ( ρ ) (17) := min M, Λ { log M : F ( ρ, Λ(Ψ M )) ≥ − ε, Λ ∈ Θ , M ∈ Z + } . We also consider a catalytic version of entanglement dilu-tion under δ -non-entangling maps. Definition 8.
The one-shot catalytic entanglement cost of ρ under a class of quantum operations Θ is defined as ˜ E (1) ,εC, Θ ( ρ ) := min M,K, Λ { log M : Λ(Ψ M ⊗ Ψ K ) = ρ ′ ⊗ Ψ K ,F ( ρ, ρ ′ ) ≥ − ε, Λ ∈ Θ , M, K ∈ Z + } . Finally, the next definition formalizes the notion of single-shot entanglement distillation under general classes of maps.
Definition 9.
The one-shot distillable entanglement of ρ undera class of quantum operations Θ is defined as E (1) ,εD, Θ ( ρ ) (18) := max M, Λ { log M : F (Λ( ρ ) , Ψ M ) ≥ − ε, Λ ∈ Θ , M ∈ Z + } . In the following we shall consider Θ to be either the classof SEPP maps or the class of δ -SEPP maps for a given δ > . The acronym comes from the name separability preserving.
III. M
AIN R ESULTS
The main results of the paper are given by the followingfour theorems. They provide operational interpretations of thesmooth max- and min-relative entropies of entanglement, andthe logarithmic version of the robustness of entanglement, interms of optimal rates of one-shot entanglement manipulationprotocols.The first theorem relates the smoothed min-relative entropyof entanglement to the single-shot distillable entanglementunder non-entangling maps.
Theorem 1.
For any state ρ and any ε ≥ , ⌊ E ε min ( ρ ) ⌋ ≤ E (1) ,εD,SEP P ( ρ ) ≤ E ε min ( ρ ) . (19)The following theorem relates the smoothed logarithmicrobustness of entanglement to the one-shot entanglement costunder non-entangling maps. Theorem 2.
For any state ρ and any ε ≥ , LR ε ( ρ ) ≤ E (1) ,εC, SEPP ( ρ ) ≤ LR ε ( ρ ) + 1 . (20)We also prove an analogous theorem to the previous one,but now relating the logarithmic global robustness (alias max-relative entropy of entanglement) to the one-shot catalyticentanglement cost under δ -non-entangling maps. Theorem 3.
For any δ, ε > there exists a positive integer K , such that for any state ρE ε max ( ρ ⊗ Ψ K ) − log K − log(1 + δ ) ≤ e E (1) ,εC,δ − SEP P ( ρ ) ≤ E ε max ( ρ ⊗ Ψ K ) − log(1 − ε ) − log K + 1 . (21) We can take in particular K = ⌈ δ − ⌉ . Finally we show that we can partially recover the reversibil-ity of entanglement manipulations under asymptotically non-entangling maps [14], [28] from the results derived in thispaper and the quantum hypothesis testing result of [19].
Theorem 4.
For every state ρ ∈ D ( H ) , E min ( ρ ) = E max ( ρ ) = E ∞ R ( ρ ) . (22)From Theorems 1 and 3 we then find that the distill-able entanglement and the catalytic entanglement cost underasymptotically non-entangling maps are the same. In Refs.[14], [28] one could show the same result without the needof catalysis. Here we need the extra resource of catalyticmaximally entangled states because we want to ensure thatalready on a single-copy level, our operations only generate anegligible amont of entanglement; in Refs. [14], [28], in turn,this is only the case for a large number of copies of the state.In more detail: we define the distillable entanglement undernon-entangling operations as E neD ( ρ ) := lim ε → lim n →∞ n E (1) ,εD,SEP P ( ρ ⊗ n ) . (23)It then follows easily from Theorem 1 and Theorem 4 that E neD ( ρ ) = E ∞ R ( ρ ) . The catalytic entanglement cost under asymptotic non-entangling operations, in turn, is defined as E aneC ( ρ ) := lim ε → lim δ → lim n →∞ n e E (1) ,εC,δ − SEP P ( ρ ) . (24)That E aneC ( ρ ) = E ∞ R ( ρ ) then follows from Theorems 3 and4. We note that it was already proven in Refs.[32], [19]that E max ( ρ ) = E ∞ R ( ρ ) . Our contribution is to show thatalso the regularization of the smooth min-relative entropy ofentanglement is equal to the regularized relative entropy ofentanglement. IV. P ROOF OF T HEOREM
Lemma 1.
For any Λ ∈ SEPP , E ε min ( ρ ) ≥ E ε min (Λ( ρ )) (25) Proof:
Let ≤ A ≤ I be such that Tr( A Λ( ρ )) ≥ − ε and E ε min (Λ( ρ )) = min σ ∈S ( − log Tr( Aσ )) . Setting σ ρ as theoptimal state in the definition of E ε min ( ρ ) , E ε min ( ρ ) ≥ − log Tr(Λ † ( A ) σ ρ )= − log Tr( A Λ( σ ρ )) ≥ min σ ∈S ( − log Tr( Aσ ))= E ε min (Λ( ρ )) . (26)where Λ † is the adjoint map of Λ . In the first line we usedthat ≤ Λ † ( A ) ≤ I and Tr(Λ † ( A ) ρ ) = Tr( A Λ( ρ )) ≥ − ε ,while in the third line we use the fact that Λ( σ ρ ) is separable,since Λ ∈ SEPP . Theorem 1:
We first prove that E (1) ,εD, SEPP ≥ ⌊ E ε min ( ρ ) ⌋ . For this it suffices to prove that any R ≤ ⌊ E ε min ( ρ ) ⌋ is anachievable one-shot distillation rate for ρ .Consider the class of completely positive trace-preservingmaps Λ ≡ Λ A (for an operator ≤ A ≤ I ) whose action ona state ρ is given as follows: Λ( ρ ) := Tr( Aρ )Ψ M + Tr (cid:0) ( I − A ) ρ (cid:1) ( I − Ψ M ) M − , (27)for any state ρ ∈ D ( H ) . An isotropic state ω , as the oneappearing on the right-hand side of Eq. (27), is separable if andonly if Tr( ω Ψ M ) ≤ /M [37]. Hence, the map Λ is SEPP if,and only if, for any separable state σ , Tr(Λ( σ )Ψ M ) ≤ /M ,or equivalently, Tr( Aσ ) ≤ M . (28)We now choose A as the optimal POVM element in thedefinition of E ε min ( ρ ) and set M = 2 ⌊ E ε min ( ρ ) ⌋ .On one hand, as Tr( Aρ ) ≥ − ε , we find that F (Λ( ρ ) , Ψ M ) ≥ − ε . On the other hand, by the definitionof E ε min ( ρ ) , we have that − E ε min ( ρ ) = max σ ∈S Tr( Aσ ) (29)and hence Tr( Aσ ) ≤ /M for every separable state σ , whichensures that the map Λ defined by (27) is a SEPP map. Hence, log M = ⌊ E ε min ( ρ ) ⌋ is an achievable rate and E (1) ,εD, SEPP ≥⌊ E ε min ( ρ ) ⌋ . We next prove the converse, namely that E (1) ,εD, SEPP ( ρ ) ≤ E ε min ( ρ ) . Suppose Λ is the optimal SEPP map such that F (Λ( ρ ) , Ψ M ) ≥ − ε , with log M = E (1) D,ε ( ρ ) .By Lemma 1 we have E ε min ( ρ ) ≥ E ε min (Λ( ρ ))= max ≤ A ≤ I Tr( A Λ( ρ )) ≥ − ε min σ ∈S ( − log Tr( Aσ )) ≥ min σ ∈S ( − log Tr(Ψ M σ ))= log M = E (1) D,ε ( ρ ) , (30)where we used that ≤ Ψ M ≤ I and Tr(Λ( ρ )Ψ M ) ≥ − ε and that Tr(Ψ M σ ) ≤ /M for every separable state σ .V. P ROOF OF T HEOREM Proof:
To prove the upper bound in (20), consider thequantum operation Λ acting on a state ω as follows: Λ( ω ) = Tr(Ψ M ω ) ρ ε + (cid:2) − Tr(Ψ M ω ) (cid:3) π, (31)where ρ ε is the state in B ε ( ρ ) which achieves the minimumin the definition (11)of the smooth logarithmic robustness, and π is a separable state such that the state σ := (cid:0) ρ ε + ( M − π (cid:1) /M, is separable for the choice M = 1 + ⌈ R ( ρ ε ) ⌉ .We can rewrite Eq. (31) as Λ( ω ) = q (cid:2) ρ ε + ( M − πM (cid:3) + (1 − q ) π, (32)where q = M Tr(Ψ M ω ) . For a separable state ω , Tr(Ψ M ω ) ≤ /M [40], and hence ≤ q ≤ . By the convexity of therobustness [39] we have that, for any separable state ω , R (Λ( ω )) ≤ qR ( σ ) + (1 − q ) R ( π ) . Note that R ( π ) = 0 since π is separable. Moreover, since R ( σ ) = 0 for M = 1 + ⌈ R ( ρ ε ) ⌉ , we have R (Λ( ω )) = 0 ,ensuring that the map Λ is non-entangling.Note that Λ(Ψ M ) = ρ ε , with the corresponding rate of log M = log(1 + ⌈ R ( ρ ε ) ⌉ ) ≤ LR ε ( ρ ) + 1 . This then yieldsthe upper bound in Theorem 2.To prove the lower bound in (20), let Λ denote a SEPP mapyielding entanglement dilution with a fidelity of at least − ε ,for a state ρ , i.e. Λ M (Ψ M ) = ρ ε , with F ( ρ, ρ ε ) ≥ − ε , and log M = E (1) ,εC, SEPP . The monotonicity of log robustness underSEPP maps [15] yields LR ε ( ρ ) ≤ LR ( ρ ε ) = LR (Λ(Ψ M )) ≤ LR (Ψ M )= log M = E (1) ,εC, SEPP . (33) VI. P
ROOF OF T HEOREM
Lemma 2.
For any δ > and Λ ∈ δ - SEPP , E ε max ( ρ ) ≥ E ε max (Λ( ρ )) − log(1 + δ ) (34) Proof:
Let ρ ε be the optimal state in the definition of E ε max ( ρ ) ,i.e., E ε max ( ρ ) = E max ( ρ ε ) . By the monotonicity of the fidelityunder CPTP maps we have that F (Λ( ρ ) , Λ( ρ ε )) ≥ F ( ρ, ρ ε ) ≥ − ε. Hence, using Lemma IV.1 of [15] E ε max (Λ( ρ )) ≤ E max (Λ( ρ ε )) ≤ E max ( ρ ε ) + log(1 + δ )= E ε max ( ρ ) + log(1 + δ ) . (35) Lemma 3.
For every ρ ∈ D ( H ) and ε > , there is a state µ ε of the form µ ε := (1 − λ ) ρ ε ⊗ Ψ K + λθ ⊗ (cid:18) I − Ψ K K − (cid:19) , (36) with K ∈ Z + } , θ, ρ ε ∈ D ( H ) , F ( ρ, ρ ε ) ≥ − ε , and λ ≤ ε ,such that E ε max ( ρ ⊗ Ψ K ) ≥ E max ( µ ε ) . (37) Proof:
Let µ ′ ε be such that E ε max ( ρ ⊗ Ψ K ) = E max ( µ ′ ε ) .Then there is a separable state σ such that µ ′ ε ≤ E ε max ( ρ ⊗ Ψ K ) σ (38)and F ( µ ′ ε , ρ ⊗ Ψ K ) ≥ − ε . Consider the twirling map ∆( X ) := Z Haar dU ( U ⊗ U ∗ ) X ( U ⊗ U ∗ ) † (39)and define µ ε := ( I ⊗ ∆)( µ ′ ε ) . Then, because ∆ is entangle-ment breaking [29] we can write µ ε := (1 − λ ) ρ ε ⊗ Ψ K + λθ ⊗ (cid:18) I − Ψ K K − (cid:19) , (40)for θ, ρ ε ∈ D ( H ) and ≤ λ ≤ . From Eq. (38), µ ε ≤ E ε max ( ρ ⊗ Ψ K ) ( I ⊗ ∆) σ. (41)Since ∆ is LOCC, ( I ⊗ ∆) σ is separable and we get E max ( µ ε ) ≤ E ε max ( ρ ⊗ Ψ K ) . Moreover, from the monotonicityof the fidelity under CPTP maps, F ( µ ε , ρ ⊗ Ψ K ) ≥ − ε . Fromthis and (36) it follows that (1 − λ ) ≥ F ( ρ, ρ ε ) ≥ − ε, and thus, λ ≤ ε . Theorem 3:
Let us start by proving the achievability part,namely that for every δ > we can find a positive integer K such that e E (1) ,εC,δ − SEPP ( ρ ) ≤ E ε max ( ρ ⊗ Ψ K ) − log(1 − ε ) − log K .From Lemma 3 we know there is a state ρ ε such that F ( ρ ε , ρ ) ≥ − ε and E max ( ρ ε ⊗ Ψ K ) ≤ E ε max ( ρ ⊗ Ψ K ) − log(1 − ε ) . This can be seen as follows: Let µ ε be a state of the form given by (36). From the definition of the max-relativeentropy of entanglement (Definition 5) it follows that µ ε ≤ E max ( µ ε ) σ ′ , ≤ E ε max ( ρ ⊗ Ψ K ) σ ′ . (42)for some separable state σ ′ ∈ B ( H ) , where we get the secondinequality by using Lemma 3. Substituting the expression (36)of µ ε we get (1 − λ ) ρ ε ⊗ Ψ K + λθ ⊗ (cid:18) I − Ψ K K − (cid:19) ≤ E ε max ( ρ ⊗ Ψ K ) σ ′ . (43)This yields, (1 − λ ) ρ ε ⊗ Ψ K ≤ E ε max ( ρ ⊗ Ψ K ) σ ′ , (44)and hence, ρ ε ⊗ Ψ K ≤ E ε max ( ρ ⊗ Ψ K ) − log(1 − λ ) σ ′ , which in turn implies that ρ ε ⊗ Ψ K ≤ E ε max ( ρ ⊗ Ψ K ) − log(1 − ε ) σ ′ , since λ ≤ ε . Therefore, for K = ⌈ δ − ⌉ and M = ⌈ K − E ε max ( ρ ⊗ Ψ K ) − log(1 − ε ) ⌉ , we can always find a state π such that (cid:0) ( ρ ε ⊗ Ψ K ) + ( M K − π (cid:1) is an unnormalizedseparable state.Define the map Λ( ω ) = (cid:2) Tr((Ψ M ⊗ Ψ K ) ω ) (cid:3)(cid:0) ρ ε ⊗ Ψ K )+ (cid:2) Tr(( I − Ψ M ⊗ Ψ K ) ω ) (cid:3) π, (45)We now show that with our choice of parameters the map Λ is δ -SEPP. First note that since for any separable state σ ∈B ( H ⊗ H ) Tr (cid:0) (Ψ M ⊗ Ψ K ) σ (cid:1) ≤ M K , we can write Λ( σ ) = p ( ρ ε ⊗ Ψ K ) + (1 − p ) π, (46)where p ≤ MK . This in turn can be written as Λ( σ ) = q (cid:2) ( ρ ε ⊗ Ψ K ) + ( M K − πM K (cid:3) + (1 − q ) π, (47)where q = pM K . Since ≤ p ≤ /M K , we have that ≤ q ≤ . Note that the first term in parenthesis in (47) isseparable, due to the choice of π . Using the convexity of theglobal robustness we then conclude that R G (Λ( σ )) ≤ R G ( π ) ,for any separable state σ .Further, from the choice of M and K it follows that R G ( π ) ≤ R G ( ρ ε ⊗ Ψ K ) ≤ K − ≤ δ. The first inequality follows from the fact that if ( ρ + sσ ) isan unnormalized separable state, then so is ( σ + (1 /s ) ρ ) , andby noting that ρ + sσ s = σ + s − ρ s − . The second inequality follows from the monotonicity of R G under LOCC [30], which implies R G ( ρ ε ⊗ Ψ K ) ≥ R G (Ψ K ) and the fact R G (Ψ K ) = K − [30]. Finally, the third is aconsequence of the choice of K .Note that for ω = Ψ M ⊗ Ψ K , Λ( ω ) = Λ(Ψ M ⊗ Ψ K ) = ρ ε ⊗ Ψ K . (48)Hence the protocol yields a state ρ ε with F ( ρ, ρ ε ) ≥ − ε and the additional maximally entangled state Ψ K which wasemployed in the start of the protocol. Its role in the protocolis to ensure that the quantum operation Λ is a δ -SEPP map forany given δ > . Since the maximally entangled states Ψ M and Ψ K were employed in the protocol and Ψ K was retrievedunchanged, the rate R = (log M + log M ′ ) − log M ′ =log M ≤ E ε max ( ρ ⊗ Ψ K ) − log K − log(1 − ε )+1 , is achievable.Next we prove the bound e E (1) , C,δ − SEPP ≥ E ε max ( ρ ) − log K − log(1 + δ ) . Let Λ be a δ -SEPP map for which Λ(Ψ M ⊗ Ψ K ) = ρ ε ⊗ Ψ K . with e E (1) ,εc,δ − SEPP = log M .Then by Lemma 2, E ε max ( ρ ⊗ Ψ K ) ≤ E max ( ρ ε ⊗ Ψ K )= E max (Λ(Ψ M ⊗ Ψ K )) ≤ E max (Ψ M ⊗ Ψ K ) + log(1 + δ )= log M + log K + log(1 + δ ) . (49)Hence log M ≥ E ε max ( ρ ⊗ Ψ K ) − log K − log(1 + δ ) . (50)VII. E QUIVALENCE WITH THE REGULARIZED RELATIVEENTROPY OF ENTANGLEMENT
In this section we prove Theorem 4. The main ingredient inthe proof is a certain generalizaton of Quantum Stein’s Lemmaproved in Refs. [16], [19] and stated below as Lemma 4 forthe special case of the separable states set.
Lemma 4.
Let ρ ∈ D ( H ) . Then(Direct part): For every ε > there exists a sequence ofPOVMs { A n , I − A n } n ∈ N such that lim n →∞ Tr(( I − A n ) ρ ⊗ n ) = 0 (51) and for every n ∈ N and ω n ∈ S ( H ⊗ n ) , − log Tr( A n ω n ) n + ε ≥ E ∞M ( ρ ) . (52) (Strong Converse): If a real number ε > and a sequenceof POVMs { A n , I − A n } n ∈ N are such that for every n ∈ N and ω n ∈ S ( H ⊗ n ) , − log(Tr( A n ω n )) n − ε ≥ E ∞M ( ρ ) , (53) then lim n →∞ Tr(( I − A n ) ρ ⊗ n ) = 1 . (54) Proof: (Theorem 4). In Refs. [16], [19], [32] it wasestablished that E max ( ρ ) = E ∞ R ( ρ ) . (55) We hence focus in showing that E min ( ρ ) ≥ E ∞ R ( ρ ) , since E min ( ρ ) ≤ E ∞ R ( ρ ) follows from Eq. (55) and the fact that E max ( ρ ) ≥ E min ( ρ ) (which in turn is a direct consequence oftheir definitions). Let ε > and { A n } be an optimal sequenceof POVMs in the direct part of Lemma 4. Then for sufficientlylarge n , Tr( ρ ⊗ n A n ) ≥ − ε and thus E ε min ( ρ ⊗ n ) ≥ min σ ∈S ( H ⊗ n ) ( − log Tr( A n σ )) ≥ n ( E ∞ R ( ρ ) − ε ) , (56)where the last inequality follows from Eq. (52). Dividing bothsides by n and taking the limit n → ∞ we get E ε min ( ρ ) ≥ E ∞ R ( ρ ) − ε. (57)Since this equation holds for every ε > , we can finally takethe limit ε → to find E min ( ρ ) ≥ E ∞ R ( ρ ) . (58)A CKNOWLEDGMENTS
The authors would like to thank Martin Plenio for manyinteresting discussions on the theme of this paper. This work ispart of the QIP-IRC supported by the European Community’sSeventh Framework Programme (FP7/2007-2013) under grantagreement number 213681, EPSRC (GR/S82176/0) as well asthe Integrated Project Qubit Applications (QAP) supported bythe IST directorate as Contract Number 015848’. FB is sup-ported by an EPSRC Postdoctoral Fellowship for TheoreticalPhysics. R
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