CCatalytic Entanglement
Tulja Varun Kondra, ∗ Chandan Datta, and Alexander Streltsov
Centre for Quantum Optical Technologies, Centre of New Technologies,University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland
Quantum entanglement of pure states is usually quan-tified via the entanglement entropy, the von Neumann en-tropy of the reduced state [1–3]. Entanglement entropyis closely related to entanglement distillation [1], a processfor converting quantum states into singlets, which can thenbe used for various quantum technological tasks. The rela-tion between entanglement entropy and entanglement dis-tillation has been known only for the asymptotic setting,and the meaning of entanglement entropy in the single-copy regime has so far remained open. Here we closethis gap by considering entanglement catalysis. We provethat entanglement entropy completely characterizes statetransformations in the presence of entangled catalysts.Our results suggest that catalysis is useful for a broadrange of quantum information protocols, giving asymp-totic results an operational meaning also in the single-copysetup.
Originated in chemistry, catalysis allows to increase the rateof a chemical reaction. This is achieved by using a catalyst,a substance which is not consumed in the process, and canthus be used repeatedly without additional costs. Similarly, aquantum catalyst is a quantum system which is not changed bythe process under consideration, giving access to transforma-tions which are not achievable without it. As has been realizedin the early days of quantum information science, catalysiscan significantly improve our abilities to transform entangledquantum systems, when it comes to transformations via localoperations and classical communication (LOCC) [4–6].One of the first examples [5] demonstrating the power ofcatalysis in quantum theory involves pure entangled statesshared by two parties, Alice and Bob. Two states, denoted by | ψ (cid:105) AB and | φ (cid:105) AB , are chosen such that no LOCC procedure canconvert | ψ (cid:105) AB into | φ (cid:105) AB . Such states can be found using con-ditions for LOCC transformations presented in [4]. Even if adirect conversion from | ψ (cid:105) AB to | φ (cid:105) AB is not possible, in somecases conversion can still be achieved by using a catalyst. Thisis an additional quantum system in an entangled state | µ (cid:105) A (cid:48) B (cid:48) ,enabling the transformation | ψ (cid:105) AB ⊗ | µ (cid:105) A (cid:48) B (cid:48) → | φ (cid:105) AB ⊗ | µ (cid:105) A (cid:48) B (cid:48) .Since the state of the catalyst remains unchanged in the pro-cess, it can be reused for another transformation in the future.A complete characterization of pure quantum states which canbe transformed into each other via LOCC with a catalyst hasso far remained open. Partial results addressing this questionhave been presented over the last decades [7–12].Catalysis is also useful in quantum thermodynamics, liftingthe well known second law in the classical domain to manysecond laws in the quantum regime [13]. Catalytic propertiesof quantum coherence [14, 15], purity [16], and theories hav- FIG. 1. Catalytic LOCC transformation from ρ S to σ S with a catalyst τ Cn . The state of the catalyst does not change in the procedure, andthe system becomes decoupled from the catalyst for n → ∞ . If ρ S and σ S are bipartite pure states, the transition is fully characterizedby entanglement entropy of the states. ing certain symmetries [17] have also been considered. Therehas also been significant interest in correlated catalysts [18–21]. Allowing a catalyst to build up correlations with the sys-tem has shown to enhance the transformation power of thecorresponding procedure [22–24].In this Letter we consider catalytic LOCC transformations .For a bipartite system S = AB a catalytic LOCC transforma-tion is defined as ρ S → lim n →∞ Tr C (cid:104) Λ n (cid:16) ρ S ⊗ τ Cn (cid:17)(cid:105) . (1)Here, C = A (cid:48) B (cid:48) is a bipartite system of the catalyst, (cid:110) τ Cn (cid:111) is asequence of catalyst states, and { Λ n } is a sequence of LOCCprotocols. We require that the catalyst is unchanged for each n , and becomes decoupled from the system in the limit n →∞ , see Fig. 1 and the methods section for more details.We will now show that for pure states catalytic LOCC trans-formations are closely related to asymptotic LOCC transfor-mations. In the asymptotic setting, the parties can operateon a large number of copies of the initial state simultane-ously. The figure of merit for the process is the transfor-mation rate, giving the maximal number of copies of | φ (cid:105) AB achievable per copy of the initial state | ψ (cid:105) AB . It has beenshown in [1] that the optimal rate is given by H (cid:16) ψ A (cid:17) / H (cid:16) φ A (cid:17) ,where ψ A and φ A are the reduced states of | ψ (cid:105) AB and | φ (cid:105) AB ,and H ( ρ ) = − Tr (cid:2) ρ log ρ (cid:3) is the von Neumann entropy. Forpure states the von Neumann entropy of the reduced state isalso a quantifier of entanglement, known as entanglement en-tropy [1–3]: E (cid:16) | ψ (cid:105) AB (cid:17) = H (cid:16) ψ A (cid:17) . If the initial and the tar-get state have the same entanglement entropy, then conversion | ψ (cid:105) AB → | φ (cid:105) AB is possible with unit rate.We are now ready to present the first result of this Letter.Alice and Bob can convert | ψ (cid:105) AB into | φ (cid:105) AB via catalytic LOCC a r X i v : . [ qu a n t - ph ] F e b FIG. 2. Catalytic quantum state merging. Alice, Bob, and Referee share a single copy of | ψ (cid:105) RAB . Alice aims to send her part of the state to Bobby using catalytic LOCC, and the Referee can apply local unitary transformations. The process is completely characterized by the quantumconditional entropy H ( A | B ), see the main text for more details. if and only if H (cid:16) ψ A (cid:17) ≥ H (cid:16) φ A (cid:17) . (2)This result means that for pure states catalysts enhance thetransformation power of LOCC, making the transformationsas powerful as in the asymptotic limit. The result in Eq. (2)is a consequence of the following theorem, concerning trans-formations from a general state ρ S into a pure state | φ (cid:105) S of ageneral multipartite system S via multipartite LOCC. Theorem 1. If ρ S can be transformed into | φ (cid:105) S via asymp-totic LOCC with unit rate, then there exists a catalytic LOCCprotocol transforming ρ S into | φ (cid:105) S . We refer to the methods section for the proof which isinspired by techniques introduced very recently in quantumthermodynamics [24]. In the bipartite setting, this theoremdirectly implies that | ψ (cid:105) AB can be transformed into | φ (cid:105) AB viacatalytic LOCC if Eq. (2) is fulfilled. As we show in themethods section by using properties of entanglement quan-tifiers [25, 26], a transformation is not possible if Eq. (2) isviolated.Remarkably, our theorem holds not only for bipartite statetransformations, but also for multipartite LOCC protocols.Here the goal is to convert a multipartite state ρ S into a purestate | φ (cid:105) S via multipartite LOCC. As a consequence, it allowsus to translate a broad range of asymptotic results in entangle-ment theory to a corresponding result on the single-copy level.We show it explicitly for a variation of quantum state merg-ing, which we term catalytic quantum state merging . Beforewe present this task, we review the standard quantum statemerging procedure in the following.In quantum state merging [27, 28], Alice, Bob, and Refereeshare asymptotically many copies of a quantum state | ψ (cid:105) RAB .The goal of the process is to send Alice’s part of the state toBob, while preserving correlations with the Referee. Aliceand Bob can perform LOCC protocols and share additionalsinglets. As was shown in [27, 28], the performance of thisprocess is characterized by the quantum conditional entropy H ( A | B ) = H (cid:16) ψ AB (cid:17) − H (cid:16) ψ B (cid:17) . (3)For H ( A | B ) > H ( A | B ), and merging is not possible if less singlets are available. For H ( A | B ) ≤ − H ( A | B ). Remarkably, quantum state merging gives an opera-tional meaning to the quantum conditional entropy, regardlesswhether H ( A | B ) is positive or negative.We are now ready to define catalytic quantum state merg-ing, giving the quantum conditional entropy an operationalmeaning also in the single-copy regime. Here, Alice, Bob,and Referee share one copy of the state | ψ (cid:105) RAB , and can useadditional catalysts in arbitrary states τ R (cid:48) A (cid:48) B (cid:48) n . While in stan-dard quantum state merging the Referee is fully inactive, incatalytic quantum state merging we allow the Referee to per-form local unitaries. However, communication between theReferee and the other parties is not required, see also Fig. 2.The goal is to merge the single copy of | ψ (cid:105) RAB on Bob’s sidewithout changing the state of the catalyst for all n , and withdecoupling of the catalyst in the limit n → ∞ . We find thatfor H ( A | B ) > H ( A | B ). This procedure isoptimal: merging is not possible if a pure state with a smallerentanglement entropy is provided. If H ( A | B ) ≤
0, then cat-alytic state merging can be performed without extra entan-glement. In the end of the process, Alice and Bob can gainan additional pure state with entanglement entropy − H ( A | B ).Also this procedure is optimal: it is not possible to achievemerging and gain a pure state with entanglement entropy ex-ceeding − H ( A | B ).As a final example we discuss assisted entanglement distil-lation [29, 30], where three parties, Alice, Bob, and Charlie,share a pure state | ψ (cid:105) ABC . By performing LOCC involvingall parties, their aim is to extract singlets between Alice andBob. In the asymptotic setup, the optimal singlet rate is givenby min (cid:110) H (cid:16) ψ A (cid:17) , H (cid:16) ψ B (cid:17)(cid:111) [30]. Correspondingly, catalytic as-sisted entanglement distillation involves one copy of | ψ (cid:105) ABC .By applying catalytic LOCC, the parties aim to establish astate | φ (cid:105) AB shared by Alice and Bob, having entanglement en-tropy as large as possible. We find that min (cid:110) H (cid:16) ψ A (cid:17) , H (cid:16) ψ B (cid:17)(cid:111) corresponds to the maximal entanglement entropy achievablefrom | ψ (cid:105) ABC in this procedure.In summary, we have shown that catalysis o ff ers a signifi-cant enhancement for entangled state transformations, leadingto e ffi ciencies previously known only for asymptotic setups.We have demonstrated this explicitly for bipartite pure statetransitions, quantum state merging, and assisted entanglementdistillation. It is reasonable to assume that similar results willhold for other quantum information protocols, including alsothe recently developed procedures for entangled state trans-formations in multipartite setups [31]. Our results suggest afull equivalence between asymptotic and catalytic entangle-ment theory. A rigorous proof of this equivalence is left openfor future research.We acknowledge financial support by the “Quantum Opti-cal Technologies” project, carried out within the InternationalResearch Agendas programme of the Foundation for PolishScience co-financed by the European Union under the Euro-pean Regional Development Fund. METHODSCatalytic LOCC
A catalytic LOCC on the system in a state ρ S is acting as inEq. (1), where (cid:110) τ Cn (cid:111) is a sequence of catalyst states and { Λ n } isa sequence of LOCC protocols. We require that for all n thecatalyst remains unchanged in the process:Tr S (cid:104) Λ n (cid:16) ρ S ⊗ τ Cn (cid:17)(cid:105) = τ Cn . (4)In general, we do not bound the dimension of the catalyst, andwe further require that the system decouples from the catalystfor large n . In particular, for a catalytic transformation from ρ S to σ S we require:lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ SCn − σ S ⊗ τ Cn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = , (5)where || M || = Tr √ M † M is the trace norm and µ SCn =Λ n (cid:16) ρ S ⊗ τ Cn (cid:17) is the total final state. We are now ready to proveTheorem 1. Proof of Theorem 1
We consider a system S consisting of m -parties and re-stricted to LOCC. Let all the parties share a state ρ . Moreover,we assume that for any ε > n and anLOCC transformation Λ such that Λ (cid:16) ρ ⊗ n (cid:17) = Γ , s.t , D (cid:16) Γ , | φ (cid:105)(cid:104) φ | ⊗ n (cid:17) < ε, (6)where D ( ρ, σ ) = || ρ − σ || is the trace distance. The above re-lation implies that | φ (cid:105) is asymptotically achievable from ρ withunit rate. We will now show that in this case there also ex-ists a catalytic LOCC procedure transforming ρ into | φ (cid:105) . Thefollowing proof is inspired by techniques introduced very re-cently within quantum thermodynamics [24]. Consider a catalyst in the state τ = n n (cid:88) k = ρ ⊗ k − ⊗ Γ n − k ⊗ | k (cid:105) (cid:104) k | . (7)The Hilbert space of the catalyst is in S ⊗ n − ⊗ K , where n isthe integer introduced in Eq. (6) and K represents an auxil-iary system of dimension n . For brevity, we denote the initialsystem S as S , and n − S , . . . , S n . Thus, thestate of the catalyst acts on S ⊗ · · · ⊗ S n ⊗ K . Moreover, Γ isa quantum state on S ⊗ S ⊗ · · · ⊗ S n , see also Eq. (6), and Γ i is the reduced state of Γ on S ⊗ S ⊗ · · · ⊗ S i . We furtherdefine Γ =
1. The auxiliary system K is maintained by Alice,serving as a register with a Hilbert space of dimension n withbasis {| k (cid:105) , k ∈ [1 , n ] } .Consider now the following LOCC protocol acting on thesystem and the catalyst:(i) Alice performs a rank-1 projective measurement on theauxiliary system K in the basis | k (cid:105) . She then communicatesthe outcome of the measurement to all the other parties. IfAlice obtains the outcome n , all parties perform the LOCCprotocol Λ given in Eq. (6) on S ⊗ S ⊗ · · · ⊗ S n . For anyother outcome of Alice’s measurement the parties do nothing.(ii) Alice applies a unitary on the auxiliary system that con-verts | n (cid:105) −→ | (cid:105) and | i (cid:105) −→ | i + (cid:105) .(iii) Finally, all the parties apply a SWAP unitary on theirparts of ( S i , S i + ) and ( S , S n ), which shifts S i −→ S i + and S n −→ S .The initial state of the system and the catalyst is given by ρ ⊗ τ = n n (cid:88) k = ρ ⊗ k ⊗ Γ n − k ⊗ | k (cid:105) (cid:104) k | . (8)After applying step (i), the initial state transforms to µ i = n n − (cid:88) k = ρ ⊗ k ⊗ Γ n − k ⊗ | k (cid:105) (cid:104) k | + n Γ ⊗ | n (cid:105) (cid:104) n | . (9)In step (ii), µ i transforms to µ ii , where µ ii = n n (cid:88) k = ρ ⊗ k − ⊗ Γ n + − k ⊗ | k (cid:105) (cid:104) k | . (10)Note that tracing out S n from µ ii gives τ , which is the initialstate of the catalyst, see Eq. (7). Therefore, using step (iii), wetransform µ ii to the final state µ having the property Tr S [ µ ] = τ . This proves that the state of the catalyst does not change inthis procedure.Now, we are left to show that Tr C [ µ ] is ε close to | φ (cid:105) . Tr C [ µ ]can be expressed as Tr C [ µ ] = n n (cid:88) k = γ k , (11)where γ k = Tr , ,..., k − , k + ,..., n [ Γ ] (12)is the reduced state of Γ on S k . Using triangle inequality andmonotonicity of the trace distance under partial trace we ob-tain D n n (cid:88) k = γ k , | φ (cid:105)(cid:104) φ | ≤ n n (cid:88) k = D ( γ k , | φ (cid:105)(cid:104) φ | ) (13) ≤ n n (cid:88) k = D (cid:16) Γ , | φ (cid:105)(cid:104) φ | ⊗ n (cid:17) < ε, where in the last inequality we used Eq. (6).The results just presented prove that if ρ can be convertedinto | φ (cid:105) with unit rate via asymptotic LOCC, then for any ε > τ and a LOCC protocol Λ suchthat σ SC = Λ (cid:16) ρ S ⊗ τ C (cid:17) , (14a) σ C = τ C , D (cid:16) σ S , | φ (cid:105)(cid:104) φ | S (cid:17) < ε. (14b)We will now show that the system and the catalyst decouplein this procedure, and moreover D (cid:16) σ SC , | φ (cid:105)(cid:104) φ | S ⊗ τ C (cid:17) < ε + √ ε. (15)In the first step, note that Eqs. (14) imply the inequality F (cid:16) σ S , | φ (cid:105)(cid:104) φ | S (cid:17) > √ − ε, (16)with fidelity F ( ρ, σ ) = Tr (cid:113) √ ρσ √ ρ . The state σ S has a pu-rification | µ (cid:105) S T = (cid:88) i λ i | i (cid:105) S | i (cid:105) T (17)with the Schmidt coe ffi cient λ i sorted in decreasing order. Dueto Eq. (16) we have λ > √ − ε. (18)Let now | ν (cid:105) SCD be a purification of σ SC , and observe that itcan be written as | ν (cid:105) SCD = (cid:88) i λ i | i (cid:105) S | α i (cid:105) CD , (19)where λ i are the same Schmidt coe ffi cients as in Eq. (17) and {| α i (cid:105)} is an orthonormal basis on CD . Noting that F (cid:16) | ν (cid:105)(cid:104) ν | SCD , | (cid:105)(cid:104) | S ⊗ | α (cid:105)(cid:104) α | CD (cid:17) = λ , (20)and using the fact that the fidelity does not decrease underpartial trace we obtain F (cid:16) σ SC , | (cid:105)(cid:104) | S ⊗ Tr D (cid:104) | α (cid:105)(cid:104) α | CD (cid:105)(cid:17) > √ − ε. (21)Using the inequality D ( ρ, σ ) ≤ (cid:112) − F ( ρ, σ ) we arrive at D (cid:16) σ SC , | (cid:105)(cid:104) | S ⊗ Tr D (cid:104) | α (cid:105)(cid:104) α | CD (cid:105)(cid:17) < √ ε. (22) Since ε can be chosen arbitrary small, this result shows that σ SC can be made arbitrary close to a product state.Noting that the trace norm does not increase under partialtrace and using Eqs. (14) and (22) we obtain D (cid:16) τ C , Tr D (cid:104) | α (cid:105)(cid:104) α | CD (cid:105)(cid:17) < √ ε, (23)where τ C is the state of the catalyst. We now use the triangleinequality, arriving at D (cid:16) σ SC , | (cid:105)(cid:104) | S ⊗ τ C (cid:17) ≤ D (cid:16) σ SC , | (cid:105)(cid:104) | S ⊗ Tr D (cid:104) | α (cid:105)(cid:104) α | CD (cid:105)(cid:17) + D (cid:16) | (cid:105)(cid:104) | S ⊗ Tr D (cid:104) | α (cid:105)(cid:104) α | CD (cid:105) , | (cid:105)(cid:104) | S ⊗ τ C (cid:17) < √ ε. (24)Using again Eq. (22) we find D (cid:16) σ S , | (cid:105)(cid:104) | S (cid:17) < √ ε, (25)which together with Eqs. (14) and triangle inequality impliesthat D (cid:16) | φ (cid:105)(cid:104) φ | S , | (cid:105)(cid:104) | S (cid:17) < ε + √ ε. (26)Using once again the triangle inequality we obtain Eq. (15): D (cid:16) σ SC , | φ (cid:105)(cid:104) φ | S ⊗ τ C (cid:17) ≤ D (cid:16) σ SC , | (cid:105)(cid:104) | S ⊗ τ C (cid:17) + D (cid:16) | (cid:105)(cid:104) | S ⊗ τ C , | φ (cid:105)(cid:104) φ | S ⊗ τ C (cid:17) < ε + √ ε. (27)This proves that the system and the catalyst decouple in theprocedure, and that Eq. (5) is fulfilled. This completes theproof of Theorem 1. Catalytic LOCC and squashed entanglement
Here we show that in the case of bipartite LOCC opera-tions squashed entanglement, introduced in [25], is monotonicunder catalytic LOCC transformations when the final state ispure. For bipartite quantum states ρ AB the squashed entangle-ment is defined as [25] E sq (cid:16) ρ AB (cid:17) = inf (cid:40) I ( A ; B | E ) : ρ ABE extension of ρ AB (cid:41) , (28)where the infimum is taken over all quantum states ρ ABE with ρ AB = Tr E (cid:16) ρ ABE (cid:17) and I ( A ; B | E ) = H (cid:16) ρ AE (cid:17) + H (cid:16) ρ BE (cid:17) − H (cid:16) ρ ABE (cid:17) − H (cid:16) ρ E (cid:17) is the quantum conditional mutual informa-tion of ρ ABE .We use the following properties of the squashed entangle-ment [25]:(a) E sq is an entanglement monotone, i.e. it does not in-crease under LOCC.(b) E sq is superadditive in general and additive on tensorproducts: E sq (cid:16) ρ AA (cid:48) BB (cid:48) (cid:17) ≥ E sq (cid:16) ρ AB (cid:17) + E sq (cid:16) ρ A (cid:48) B (cid:48) (cid:17) (29)and equality holds true if ρ AA (cid:48) BB (cid:48) = ρ AB ⊗ ρ A (cid:48) B (cid:48) .(c) For a pure state | ψ (cid:105) AB squashed entanglement is equalto the entanglement entropy, i.e., the entropy of the reducedstate: E sq (cid:16) | ψ (cid:105) AB (cid:17) = H (cid:16) ψ A (cid:17) . (30)(d) Squashed entanglement is continuous in the vicinity ofany pure state.We are now ready to prove the following theorem. Theorem 2.
If a bipartite state ρ AB can be transformed intothe pure state | φ (cid:105) AB via catalytic LOCC, thenE sq (cid:16) ρ AB (cid:17) ≥ E sq (cid:16) | φ (cid:105) AB (cid:17) . (31) Proof.
Assume that for any ε > τ A (cid:48) B (cid:48) and an LOCC protocol Λ such that the final state σ AA (cid:48) BB (cid:48) = Λ ( ρ AB ⊗ τ A (cid:48) B (cid:48) ) has the properties D (cid:16) Tr A (cid:48) B (cid:48) (cid:104) σ AA (cid:48) BB (cid:48) (cid:105) , | φ (cid:105)(cid:104) φ | AB (cid:17) < ε, Tr AB (cid:104) σ AA (cid:48) BB (cid:48) (cid:105) = τ A (cid:48) B (cid:48) . Using the properties (a) and (b) of the squashed entanglement,we find E sq (cid:16) σ AA (cid:48) BB (cid:48) (cid:17) ≤ E sq (cid:16) ρ AB (cid:17) + E sq (cid:16) τ A (cid:48) B (cid:48) (cid:17) (32)and also E sq (cid:16) σ AA (cid:48) BB (cid:48) (cid:17) ≥ E sq (cid:16) Tr A (cid:48) B (cid:48) (cid:104) σ AA (cid:48) BB (cid:48) (cid:105)(cid:17) + E sq (cid:16) τ A (cid:48) B (cid:48) (cid:17) . (33)From Eqs. (32) and (33) it follows E sq (cid:16) ρ AB (cid:17) ≥ E sq (cid:16) Tr A (cid:48) B (cid:48) (cid:104) σ AA (cid:48) BB (cid:48) (cid:105)(cid:17) . (34)If Tr A (cid:48) B (cid:48) (cid:104) σ AA (cid:48) BB (cid:48) (cid:105) can be made arbitrarily close to | φ (cid:105) (cid:104) φ | AB in trace distance, then using the property (d) of the squashedentanglement we get E sq (cid:16) ρ AB (cid:17) ≥ E sq (cid:16) | φ (cid:105) AB (cid:17) , and the proof iscomplete. (cid:3) Combining Theorems 1 and 2, we conclude that a pure state | ψ (cid:105) AB can be transformed into another pure state | φ (cid:105) AB via cat-alytic LOCC if and only if H (cid:16) ψ A (cid:17) ≥ H (cid:16) φ A (cid:17) , as claimed in themain text. This gives an operational interpretation for the vonNeumann entropy in the single copy scenario. Catalytic LOCC protocols in tripartite setups
We will now consider a tripartite setup, developing toolswhich will serve as a basis for catalytic quantum state merg-ing. Similar to the state merging setup [27, 28], we considerthree parties (Alice, Bob and Referee) sharing a tripartite state ρ = ρ RAB . Assume now that by applying asymptotic LOCCbetween Alice and Bob, it is possible to asymptotically con-vert ρ into pure states | φ (cid:105) = | φ (cid:105) RAB . The following theorem es-tablishes a connection between this setup and catalytic LOCC.
Theorem 3. If ρ can be converted into | φ (cid:105) via asymptoticLOCC between Alice and Bob with unit rate, then ρ can beconverted into | φ (cid:105) by applying catalytic LOCC between Aliceand Bob and a unitary on Referee’s side.Proof. The proof follows similar reasoning as the proof ofTheorem 1. If ρ can be converted into | φ (cid:105) via asymptoticLOCC between Alice and Bob with unit rate, then for any ε > n and an LOCC protocol Λ be-tween Alice and Bob such that Λ (cid:104) ρ ⊗ n (cid:105) = Γ and D (cid:16) Γ , | φ (cid:105)(cid:104) φ | ⊗ n (cid:17) < ε. (35)We now consider a catalyst in the state τ = n n (cid:88) k = ρ ⊗ k − ⊗ Γ n − k ⊗ | k (cid:105) (cid:104) k | . (36)Also in this case the Hilbert space of the catalyst is in S ⊗ n − ⊗ K , where S = RAB now corresponds to the tripartite systemof Alice, Bob and Referee. Again, we denote the n copies ofthe same systems as S , . . . , S n , where S corresponds to thesystem S , and the state of the catalyst is acting on S ⊗ · · · ⊗ S n ⊗ K . The operator Γ acts on S ⊗ S ⊗ · · · ⊗ S n and Γ i (i ∈ { , . . . , n } ) is the reduced state of Γ on S ⊗ S ⊗ · · · ⊗ S i .Moreover, K is a register on Alice’s side.We now follow a procedure very similar to the one in theproof of Theorem 1.(i) Alice performs a rank-1 projective measurement on theregister K in the basis | k (cid:105) . She then communicates the out-come of the measurement to Bob. If Alice obtains the out-come n , Alice and Bob perform the LOCC protocol Λ givenin Eq. (35). For any other outcome of Alice’s measurementthe parties do nothing.(ii) Alice applies a unitary on the auxiliary system, that con-vert | n (cid:105) −→ | (cid:105) and | i (cid:105) −→ | i + (cid:105) .(iii) Alice, Bob and Referee apply a SWAP unitary on theirparts of ( S i , S i + ) and ( S , S n ) that shift S i −→ S i + and S n −→ S . Note that communication with the Referee is notnecessary, the Referee applies the SWAP operations indepen-dently of the procedure performed by Alice and Bob.By the same reasoning as in the proof of Theorem 1, wesee that the first subsystem S of the final state is ε close to | φ (cid:105) while the catalyst remains unchanged. Moreover, the sub-system S decouples from the catalyst in the limit n → ∞ ,which is proven exactly in the same way as in Theorem 1, seealso Eq. (15). (cid:3) Catalytic quantum state merging
In quantum state merging (QSM) [27, 28], we assume thatAlice, Bob and Referee share asymptotically many copiesof a pure quantum state | ψ (cid:105) RAB . By applying LOCC opera-tions, Alice and Bob aim to transfer the state of Alice to Bobwhile preserving correlations with Referee, i.e., the final state | ψ (cid:105) RBB (cid:48) is the same as | ψ (cid:105) RAB up to relabelling of A and B (cid:48) . Inthe following, we describe three possible scenarios.a) In the asymptotic limit where many copies of the state | ψ (cid:105) RAB are available, QSM is possible if the conditional en-tropy is zero [27, 28], i.e., H ψ ( A | B ) = H (cid:16) ψ AB (cid:17) − H (cid:16) ψ B (cid:17) = . (37)This means, if H ψ ( A | B ) =
0, there exists an LOCC protocoltaking | ψ (cid:105) RAB arbitrarily close to | ψ (cid:105) RBB (cid:48) (cid:16) | ψ (cid:105) RAB (cid:17) ⊗ n ε −−−−→ LOCC (cid:16) | ψ (cid:105) RBB (cid:48) (cid:17) ⊗ n , (38)where ε above the arrow represents that the final state is ε close in trace distance to (cid:16) | ψ (cid:105) RBB (cid:48) (cid:17) ⊗ n .b) If H ψ ( A | B ) >
0, then we define | ψ (cid:48) (cid:105) RA ˜ AB ˜ B = | ψ (cid:105) RAB ⊗| φ (cid:105) ˜ A ˜ B , where | φ (cid:105) ˜ A ˜ B is a shared entangled state between Aliceand Bob with entanglement entropy E (cid:16) | φ (cid:105) ˜ A ˜ B (cid:17) = H ψ ( A | B ).This implies, H ψ (cid:48) (cid:16) A ˜ A | B ˜ B (cid:17) =
0. Therefore, from Eq. (38)we see that Alice and Bob can successfully merge the state | ψ (cid:48) (cid:105) RA ˜ AB ˜ B : (cid:16) | ψ (cid:48) (cid:105) RA ˜ AB ˜ B (cid:17) ⊗ n ε −−−−→ LOCC (cid:16) | ψ (cid:105) RBB (cid:48) (cid:17) ⊗ n . (39)c) If H ψ ( A | B ) <
0, the following transformation is achiev-able via LOCC between Alice and Bob [27, 28]: (cid:16) | ψ (cid:105) RAB (cid:17) ⊗ n ε −−−−→ LOCC (cid:16) | ψ (cid:105) RBB (cid:48) (cid:17) ⊗ n ⊗ (cid:18) | φ + (cid:105) ˜ A ˜ B (cid:19) ⊗− H ψ ( A | B ) n , (40)with the Bell state | φ + (cid:105) = ( | (cid:105) | (cid:105) + | (cid:105) | (cid:105) ) / √
2. Additionally,we know that [1] (cid:18) | φ + (cid:105) ˜ A ˜ B (cid:19) ⊗− H ψ ( A | B ) n ε −−−−→ LOCC (cid:16) | φ (cid:105) ˜ A ˜ B (cid:17) ⊗ n , (41)where | φ (cid:105) ˜ A ˜ B is a bipartite pure state with entanglement en-tropy E (cid:16) | φ (cid:105) ˜ A ˜ B (cid:17) = − H ψ ( A | B ). Therefore, from Eqs. (40)and (41) we see that the following transformation is achiev-able via LOCC between Alice and Bob: (cid:16) | ψ (cid:105) RAB (cid:17) ⊗ n ε −−−−→ LOCC (cid:16) | ψ (cid:105) RBB (cid:48) ⊗ | φ (cid:105) ˜ A ˜ B (cid:17) ⊗ n . (42)Equipped with these tools we are now ready to discuss cat-alytic quantum state merging. Recall that in catalytic QSM,we allow LOCC operations between Alice and Bob, and theReferee can also perform local unitaries. However, no com-munication between the Referee and the other parties is al-lowed. This is exactly the setup considered in Theorem 3, andwe will make use of this result in the following.An immediate consequence of Theorem 3 and Eq. (42) isthat for H ψ ( A | B ) < | φ (cid:105) ˜ A ˜ B with entanglement entropy E ( | φ (cid:105) ˜ A ˜ B ) = − H ψ ( A | B ). In the following, we prove by contradiction that this is theoptimal value one can achieve. Let us assume that there existsa catalytic LOCC procedure such that E (cid:16) | φ (cid:105) ˜ A ˜ B (cid:17) > H (cid:16) ψ B (cid:17) − H (cid:16) ψ AB (cid:17) . (43)Consider now the squashed entanglement [25, 26] betweenBob and the rest of the system in the initial state | ψ (cid:105) RAB : E B | ARsq (cid:16) | ψ (cid:105) RAB (cid:17) = H (cid:16) ψ B (cid:17) . (44)On the other hand, the squashed entanglement between Boband the rest of the system in the target state | ψ (cid:105) RBB (cid:48) ⊗ | φ (cid:105) ˜ A ˜ B isgiven by E BB (cid:48) ˜ B | ˜ ARsq (cid:16) | ψ (cid:105) RBB (cid:48) ⊗ | φ (cid:105) ˜ A ˜ B (cid:17) = H (cid:16) ψ AB (cid:17) + E (cid:16) | φ (cid:105) ˜ A ˜ B (cid:17) . (45)From Eqs. (43), (44), and (45), we obtain E BB (cid:48) ˜ B | ˜ ARsq (cid:16) | ψ (cid:105) RBB (cid:48) ⊗ | φ (cid:105) ˜ A ˜ B (cid:17) > H (cid:16) ψ B (cid:17) = E B | ARsq ( | ψ (cid:105) RAB ) . (46)Recalling that squashed entanglement is continuous in thevicinity of pure states [25], this means that the squashed en-tanglement has increased in the process. This is a contradic-tion to Theorem 2, showing that for pure states squashed en-tanglement cannot increase under catalytic LOCC.In the remaining case H ψ ( A | B ) ≥
0, from Theorem 3 andEq. (39) it directly follows catalytic QSM is possible whenAlice and Bob are provided with an additional state | φ (cid:105) ˜ A ˜ B with entanglement entropy E (cid:16) | φ (cid:105) ˜ A ˜ B (cid:17) = H ψ ( A | B ). We nowshow that this is the minimal entanglement entropy needed toperform catalytic QSM. Again we use the properties of thesquashed entanglement to prove this. The squashed entan-glement between Bob and the other parties in the initial state | ψ (cid:105) RAB ⊗ | φ (cid:105) ˜ A ˜ B is given by E B ˜ B | A ˜ ARsq (cid:16) | ψ (cid:105) RAB ⊗ | φ (cid:105) ˜ A ˜ B (cid:17) = H (cid:16) ψ B (cid:17) + E (cid:16) | φ (cid:105) ˜ A ˜ B (cid:17) . (47)For the target state | ψ (cid:105) RBB (cid:48) we obtain E BB (cid:48) | Rsq (cid:16) | ψ (cid:105) RBB (cid:48) (cid:17) = H (cid:16) ψ AB (cid:17) . (48)Using again the fact that squashed entanglement is continu-ous in the vicinity of pure states and cannot increase undercatalytic LOCC, we have E B ˜ B | A ˜ ARsq (cid:16) | ψ (cid:105) RAB ⊗ | φ (cid:105) ˜ A ˜ B (cid:17) ≥ E BB (cid:48) | Rsq (cid:16) | ψ (cid:105) RBB (cid:48) (cid:17) . (49)Hence, from Eqs. (47), (48) and (49), we get an achievablelower bound on the entanglement entropy of | φ (cid:105) ˜ A ˜ B : E (cid:16) | φ (cid:105) ˜ A ˜ B (cid:17) ≥ H (cid:16) ψ AB (cid:17) − H (cid:16) ψ B (cid:17) . (50) Catalytic assisted entanglement distillation
Consider now three parties, Alice, Bob, and Charlie, shar-ing a pure state | ψ (cid:105) ABC . By performing catalytic LOCC be-tween all the parties, they aim to convert | ψ (cid:105) ABC into a state | φ (cid:105) AB which has maximal possible entanglement entropy. Thistask is analogous to assisted entanglement distillation, whichhas been previously studied in the asymptotic setting [30].We will now show that the optimal procedure is for Charlieto merge his state either with Alice or with Bob. For this,consider the corresponding conditional mutual information H ψ ( C | A ) = H (cid:16) ψ AC (cid:17) − H (cid:16) ψ A (cid:17) = H (cid:16) ψ B (cid:17) − H (cid:16) ψ A (cid:17) , (51) H ψ ( C | B ) = H (cid:16) ψ BC (cid:17) − H (cid:16) ψ B (cid:17) = H (cid:16) ψ A (cid:17) − H (cid:16) ψ B (cid:17) . (52)We immediately see that either H ψ ( C | A ) and H ψ ( C | B ) are bothzero, or at least one of them is negative.If H ψ ( C | A ) <
0, then Charlie merges his system with Aliceby using catalytic QSM. As a result, Alice and Bob will endup with a state having entanglement entropy H (cid:16) ψ B (cid:17) . Note that H ψ ( C | A ) < H (cid:16) ψ B (cid:17) < H (cid:16) ψ A (cid:17) . On the otherhand, if H ψ ( C | B ) ≤
0, then Charlie merges his system withBob, leaving Alice and Bob with a state having entanglemententropy H (cid:16) ψ A (cid:17) . Since H ψ ( C | B ) ≤ H (cid:16) ψ A (cid:17) ≤ H (cid:16) ψ B (cid:17) , this proves that via catalytic LOCC it is possible toconvert | ψ (cid:105) RAB into a quantum state | φ (cid:105) AB having entanglemententropy E (cid:16) | φ (cid:105) AB (cid:17) = min (cid:110) H (cid:16) ψ A (cid:17) , H (cid:16) ψ B (cid:17)(cid:111) . (53)The converse can be proven by using the properties of thesquashed entanglement [25], in particular that for pure statesit corresponds to the entanglement entropy and does not in-crease under catalytic LOCC, see Theorem 2. Since any tri-partite LOCC protocol is also bipartite with respect to any bi-partition, it must be that E A | BRsq (cid:16) | ψ (cid:105) RAB (cid:17) ≥ E A | Bsq (cid:16) | φ (cid:105) AB (cid:17) , (54) E B | ARsq (cid:16) | ψ (cid:105) RAB (cid:17) ≥ E A | Bsq (cid:16) | φ (cid:105) AB (cid:17) . 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