PPT-inducing, distillation-prohibiting, and entanglement-binding quantum channels
PPPT-inducing, distillation-prohibiting, and entanglement-binding quantum channels
Sergey N. Filippov
Moscow Institute of Physics and Technology, Institutskii Per. 9, Dolgoprudny, Moscow Region 141700, RussiaInstitute of Physics and Technology, Russian Academy of Sciences, Nakhimovskii Pr. 34, Moscow 117218, RussiaRussian Quantum Center, Novaya 100, Skolkovo, Moscow Region 143025, Russia andP. N. Lebedev Physical Institute, Russian Academy of Sciences, Leninskii Pr. 53, Moscow 119991, Russia
Entanglement degradation in open quantum systems is reviewed in the Choi-Jamio(cid:32)lkowski rep-resentation of linear maps. In addition to physical processes of entanglement dissociation andentanglement annihilation, we consider quantum dynamics transforming arbitrary input states intothose that remain positive under partial transpose (PPT-inducing channels). Such evolutions forma convex subset of distillation-prohibiting channels. A relation between the above channels andentanglement-binding ones is clarified. An example of the distillation-prohibiting map Φ ⊗ Φ isgiven, where Φ is not entanglement binding.
PACS numbers: 03.67.Mn, 03.65.Ud, 03.65.Yz
I. INTRODUCTION
The phenomenon of quantum entanglement usuallyemerges between interacting subsystems in a compos-ite system. The long-distance and long-living forms ofsuch correlations play a vital role in up-to-date quan-tum technologies [1]. Unavoidable interaction with theenvironment changes the structure of entanglement [2].In special cases, the global environment bath can createquantum correlations between the particles of the com-posite system [3], however, local noises degrade it and im-pose limitations on achievable entanglement death timein experiments [4, 5]. Bound entangled states are thosethat are still entangled but cannot be distilled into max-imally entangled qubit pairs [6]. Positive partial trans-pose (PPT) states [7] are known to be undistillable, so itis reasonable to characterize quantum channels resultingin PPT states regardless of the input.To describe the stages of gradual entanglement degra-dation, the notions of entanglement annihilation [8,9] and entanglement dissociation [10] were introducedrecently. PPT-inducing and distillation-prohibitingchannels can be considered as their generalizationsto corresponding entanglement properties (PPT andundistillability). The previously introduced notion ofentanglement-binding channel [11] turns out to be apartial case (one-sided realization) of the distillation-prohibiting channel.The paper is organized as follows.In Sec. II, the description of quantum entangle-ment is briefly reviewed. In Sec. III, the basic in-formation about quantum channels and their entan-glement degradation properties is given. In Sec. IV,the Choi-Jamio(cid:32)lkowski representation [12–14] of linearmaps and their concatenations is discussed, the structureof Choi matrix is reviewed for entanglement-breaking,entanglement-binding, entanglement-annihilating, andentanglement-dissociating channels. Main results arepresented in Sec. V, where properties of PPT-inducingand distillation-prohibiting channels are studied. InSec. VI, brief conclusions are given.
II. QUANTUM ENTANGLEMENT
A quantum state is described by a density opera-tor (cid:37) , which is Hermitian, positive semidefinite, andhas unit trace. In a composite system, there are sev-eral degrees of freedom, say,
A, B, C, . . . , and the corre-sponding Hilbert space has the tensor product structure H A ⊗H B ⊗H C ⊗· · · . Specification of particular degrees offreedom fixes the partitioning structure A | B | C | · · · andthe associated entanglement structure [37]. In experi-ments and applications, one deals with the accessible de-grees of freedom that are naturally related to the usedmeasurement techniques. From now on, these degrees offreedom are supposed to be fixed.The state (cid:37) ABC ··· is called fully separable if there ex-ist a probability distribution { p k } and density operators (cid:37) Ak , (cid:37) Bk , (cid:37) Ck , . . . such that (cid:37) ABC ··· belongs to the closure ofthe states (cid:80) k p k (cid:37) Ak ⊗ (cid:37) Bk ⊗ (cid:37) Ck ⊗ · · · ; otherwise, (cid:37) ABC ··· is called entangled [16].An example of a coarse-grained partition is P = AB | C | DEF . The state (cid:37)
ABCDEF is separable with re-spect to P if (cid:37) ABCDEF = (cid:80) k p k (cid:37) ABk ⊗ (cid:37) Ck ⊗ (cid:37) DEFk . A fullyseparable state is separable with respect to AB | C | DEF but the converse does not hold in general. It is instruc-tive to remind about the existence of a three-qubit state (cid:37)
ABC , which is separable with respect to all bipartitions( A | BC , B | AC , and C | AB ) but is not fully separable, i.e.entangled with respect to tripartition A | B | C [17, 18].A state, which cannot be written as a convex sum ofany separable bipartite states, is usually referred to asgenuinely entangled. Under the action of local noises,the genuine entanglement degrades at first to the bisep-arable form (convex sum of states separable with respectto bipartitions), then to the triseparable form (convexsum of states separable with respect to tripartitions),and so on. Such a process is called entanglement dis-sociation [10] in analogy to the dissociation of chemicalcompounds in solvents. The final stage of entanglementevolution is annihilation, when the system state becomesfully separable.A bipartite state (cid:37) AB is called positive under partial a r X i v : . [ qu a n t - ph ] S e p transpose (PPT) if (cid:37) A ( B ) (cid:62) ≡ Id A ⊗ T B [ (cid:37) AB ] is positivesemidefinite [7]. Here, Id is the identity transformationand T B is the transposition in some orthonormal basisin H B . Clearly, (cid:37) A ( B ) (cid:62) ≥ ⇐⇒ (cid:37) ( A ) (cid:62) B ≥ (cid:37) ( AB ) (cid:62) is a valid density operator and T = Id. A sepa-rable state (cid:37) AB is necessarily PPT [7], the converse holdsif d A d B ≤ III. QUANTUM CHANNELS
Evolution of an open quantum system during the timeinterval (0 , T ) can be considered as an input-output rela-tion (cid:37) t = T = Φ[ (cid:37) t =0 ] between the final and initial densitymatrices. If the system is decoupled from the environ-ment at time t = 0, then Φ is a single-valued linear map.Physical reasoning leads to a conclusion that Φ is a com-pletely positive trace preserving map [21, 22], which wewill refer to as quantum channel (see, e.g., [23]).Suppose a channel Φ acting on the system S = ABC . . . and the initial state (cid:37) in . If the output state (cid:37) out = Φ[ (cid:37) in ] is separable with respect to a fixed partition P for all (cid:37) in , then Φ is called entanglement-dissociatingwith respect to P . If P = A | B | C | · · · , i.e. (cid:37) out isfully separable for all input states, then Φ is calledentanglement-annihilating [10].Notions of entanglement dissociation and annihila-tion do not imply the use of any auxiliary systems.In contrast, the so called entanglement breaking chan-nels are those for which Φ ⊗ Id[ (cid:37) S +ancin ] is separablewith respect to partition S | anc for any initial densityoperator (cid:37) S +ancin , with the dimension of an ancillaryHilbert space being arbitrary [24–26]. Equivalently, Φ S is entanglement-breaking if Φ S ⊗ Id anc is entanglementdissociating with respect to the bipartition S | anc for alldimensions dim H anc = 2 , , . . . .Similarly, if Φ ⊗ Id[ (cid:37) S +ancin ] is undistillable (with respectto the system S and the ancillary system) for any initialdensity operator (cid:37) S +ancin , then Φ is called entanglementbinding [11]. IV. CHOI-JAMIO(cid:32)LKOWSKIREPRESENTATION
In case of finite dimensions, a linear map Φ actingon a system S can be defined via the so-called Choi-Jamio(cid:32)lkowski isomorphism [12–14]:Ω SS (cid:48) Φ = Φ S ⊗ Id S (cid:48) [ | Ψ SS (cid:48) + (cid:105)(cid:104) Ψ SS (cid:48) + | ] , (1)Φ[ X ] = d S tr S (cid:48) [ Ω SS (cid:48) Φ ( I S out ⊗ X T ) ] , (2)where d S = dim H S = dim H S (cid:48) , | Ψ SS (cid:48) + (cid:105) =( d S ) − / (cid:80) d S i =1 | i ⊗ i (cid:48) (cid:105) is a maximally entangled stateshared by system S and its clone S (cid:48) , tr S (cid:48) denotes the partial trace over S (cid:48) , I is the identity operator, and X T = (cid:80) i,j (cid:104) j | X | i (cid:105)| i (cid:48) (cid:105)(cid:104) j (cid:48) | is the transposition in some or-thonormal basis. The operator (1) is referred to as Choioperator or Choi state (see Fig. 1). A. Review of properties
The well known results are as follows:1. Φ is positive if and only if Ω SS (cid:48) Φ is block-positive,i.e. (cid:104) ϕ S ⊗ χ S (cid:48) | Ω SS (cid:48) Φ | ϕ S ⊗ χ S (cid:48) (cid:105) ≥ ϕ, χ [13].2. Φ is completely positive (quantum operation) if andonly if Ω SS (cid:48) Φ ≥ SS (cid:48) Φ ≥ S | S (cid:48) [25].4. Φ is entanglement binding if and only if Ω SS (cid:48) Φ ≥ S and S (cid:48) ) [11].A feature of entanglement breaking maps is that theiroutcome is separable not only for positive inputs (den-sity operators) (cid:37) S +anc but also for block-positive inputs ξ S | ancBP . Since tr[Ω pos Ω ent − br ] ≥
0, the cone of entangle-ment breaking maps is dual to the cone of positive maps(see, e.g., [27, 28]).As far as multipartite composite systems S = ABC . . . are concerned, the maximally entangled statecan be viewed as separable with respect to thepartition AA (cid:48) | BB (cid:48) | CC (cid:48) | . . . , namely, | Ψ SS (cid:48) + (cid:105) =( d A d B d C · · · ) − / (cid:80) d A i =1 (cid:80) d B j =1 (cid:80) d C k =1 (cid:80) ··· | ijk · · ·(cid:105) ⊗| i (cid:48) j (cid:48) k (cid:48) · · ·(cid:105) = | Ψ AA (cid:48) + (cid:105) ⊗ | Ψ BB (cid:48) + (cid:105) ⊗ | Ψ CC (cid:48) + (cid:105) ⊗ · · · .If Φ is a local map, i.e. has the form Φ A ⊗ Φ B ⊗ Φ C ⊗ · · · , then the composite Choi operator readsΩ ABC...A (cid:48) B (cid:48) C (cid:48) ... Φ ⊗ Φ ⊗ Φ ⊗ ... = Ω AA (cid:48) Φ ⊗ Ω BB (cid:48) Φ ⊗ Ω CC (cid:48) Φ ⊗· · · . Positivity ofthis operator is equivalent to positivity of individual Choioperators. By property 2, Φ A ⊗ Φ B ⊗ Φ C ⊗ · · · is com-pletely positive if and only if each of the maps Φ A , Φ B ,Φ C , . . . is completely positive. Similarly, by properties 3and 4, Φ A ⊗ Φ B ⊗ Φ C ⊗· · · is entanglement breaking (bind-ing) if and only if each of the maps Φ A , Φ B , Φ C , . . . isentanglement breaking (binding). Nonetheless, property1 cannot be extended in analogous way. In fact, the mapΦ A ⊗ Φ B is positive if and only if Ω AA (cid:48) Φ ⊗ Ω BB (cid:48) Φ = ξ AB | A (cid:48) B (cid:48) BP ,i.e. the composite Choi matrix is block positive with re-spect to partition AB | A (cid:48) B (cid:48) .Suppose a positive map Υ acting on a composite sys-tem ABC . . . . Υ dissociates entanglement with respectto some partition P ( ABC . . . ) if and only iftr (cid:104) Ω ABC...A (cid:48) B (cid:48) C (cid:48) ... Υ (cid:16) ξ P ( ABC... )BP ⊗ (cid:37) A (cid:48) B (cid:48) C (cid:48) ... (cid:17)(cid:105) ≥ ξ P ( ABC... )BP and density opera-tors (cid:37) A (cid:48) B (cid:48) C (cid:48) ... . This result follows immediately fromEq. (2) [10]. However, in order to describe a valid (cid:2)(cid:2) (cid:3) S Choistate
S SS’ S’ (cid:3) S S SS’ S’ AA’ BB’ AA’ BB’ (cid:3)(cid:4)(cid:4)(cid:4)(cid:4)(cid:3) (cid:2)
A B (cid:3) AB AA’ BB’ AA’ BB’ A B A BA’ B’ A’ B’A B A BA’ B’ A’ B’ AA’ BB’AA’ BB’ TT (a) (b) (c) (d)(e) (f ) (g) (h) FIG. 1: Choi operator Ω SS (cid:48) Φ as a result of applying Φ S to one side of the state | Ψ SS (cid:48) + (cid:105) maximally entangled between system S and its clone S (cid:48) (a). Choi operator for a map Φ AB acting on a composite system AB (b). Visualization of property 5 forChoi matrices of entanglement-annihilating channels (c). Visualization of Corollary 1 for PPT-inducing channels (d). Choistate of entanglement-breaking channel is separable, Choi state of entanglement binding channel is undistillable (e). Choistate of local entanglement-breaking channel Φ ⊗ Φ is fully separable (f). Visualization of property 6 — sufficient conditionof entanglement annihilation — in terms of Choi matrices (g). Visualization of Proposition 2 depicting the Choi matrix ofPPT-inducing channel Φ A ⊗ Id B (h). quantum evolution, Υ has to be completely positive, i.e.Ω ABC...A (cid:48) B (cid:48) C (cid:48) ... Υ ≥
0. Choi matrices that satisfy this con-dition and requirement (3) are precisely entanglementdissociating channels with respect to the partition P .The bipartite setting of Eq. (3) readstr[Ω ABA (cid:48) B (cid:48) Φ ( ξ A | B BP ⊗ (cid:37) A (cid:48) B (cid:48) )] ≥ AB . The last formulashows that a cone of entanglement annihilatingmaps is dual to the cone of maps Θ AB of the formΘ[ X ] = (cid:80) k tr[ F k X ] ξ A | B BP k , F k ≥ † is called dual to the map Φ if tr (cid:2) Φ † [ X ] Y (cid:3) ≡ tr [ X Φ[ Y ]] for X, Y from corresponding domains[38]. Anobservation for the entanglement annihilating map Φ AB is that Φ † transforms all block-positive operators ξ A | B BP into positive ones ( ∝ (cid:37) AB ). As a consequence, the con-catenation Φ ◦ Φ † has to map block-positive operators toseparable ones (this is a necessary condition for Φ AB tobe entanglement annihilating).Finally, sufficient conditions for entanglement annihi-lation are as follows [29]:5. If Ω ABA (cid:48) B (cid:48) Φ can be represented in the form of a con-vex sum of operators ξ A | A (cid:48) B (cid:48) BP ⊗ (cid:37) B and (cid:37) A ⊗ ξ B | A (cid:48) B (cid:48) BP ,then Φ AB annihilates entanglement.6. If Ω ABA (cid:48) B (cid:48) Φ is a convex sum of separable states ofthe form (cid:37) A | BA (cid:48) B (cid:48) and (cid:37) B | AA (cid:48) B (cid:48) , then Φ AB is en-tanglement annihilating.Some key properties of Choi operators for the discussedmaps are depicted schematically in Fig. 1. B. Concatenation of maps in terms of Choioperators
A concatenation Φ ◦ Ξ of two maps Φ : M d (cid:55)→ M d andΞ : M d (cid:55)→ M d is a map such that Φ ◦ Ξ[ X ] ≡ Φ [Ξ[ X ]].It is not hard to see thatΩ Φ ◦ Ξ = d d (cid:88) i,j,k,l,m,n =1 | m ⊗ k (cid:105)(cid:104) m ⊗ i | Ω Φ | n ⊗ j (cid:105)(cid:104) i ⊗ k | Ω Ξ | j ⊗ l (cid:105)(cid:104) n ⊗ l | . (4)The rule (4) can be treated as a star-productscheme [30, 31], where the symbols are elements ofChoi matrices. The kernel of the star product reads K ( mk, nl ; pq, rs ; tu, vw ) = δ mp δ qt δ rn δ sv δ ku δ wl . V. PPT-INDUCING ANDDISTILLATION-PROHIBITING CHANNELS
A map Φ AB transforming operators acting on H A ⊗H B is called PPT-inducing if Φ AB [ (cid:37) AB ] is positive and PPTwith respect to the partition A | B for all input states (cid:37) AB .If in addition Φ AB is completely positive and trace pre-serving, then Φ AB is the PPT-inducing quantum channel. Proposition 1.
A positive map Φ AB is PPT-inducingif and only if Ω A ( B ) (cid:62) A (cid:48) B (cid:48) Φ is block-positive with respect tothe partition AB | A (cid:48) B (cid:48) .Proof. Partial transposition of formula (2) yields (cid:37) A ( B ) (cid:62) out = tr A (cid:48) B (cid:48) (cid:104) Ω A ( B ) (cid:62) A (cid:48) B (cid:48) Φ ( I AB ⊗ (cid:37) ( AB ) (cid:62) in ) (cid:105) . Pos-itivity of (cid:37) A ( B ) (cid:62) out means tr[ (cid:37) A ( B ) (cid:62) out (cid:101) (cid:37) AB ] ≥ (cid:101) (cid:37) AB , which is equivalent totr (cid:104) Ω A ( B ) (cid:62) A (cid:48) B (cid:48) Φ ( (cid:101) (cid:37) AB ⊗ (cid:37) ( AB ) (cid:62) in ) (cid:105) ≥ (cid:101) (cid:37) AB and (cid:37) AB in , i.e. block-positivity of Ω A ( B ) (cid:62) A (cid:48) B (cid:48) Φ with respect tothe partition AB | A (cid:48) B (cid:48) .Apparently, one could use Ω ( A ) (cid:62) BA (cid:48) B (cid:48) Φ instead ofΩ A ( B ) (cid:62) A (cid:48) B (cid:48) Φ in the formulation of Proposition 1. Besides,the positivity of map Φ AB is essential.Since positive operators are automatically block-positive, we readily obtain the following result. Corollary 1.
Suppose a channel Φ AB such that Ω A ( B ) (cid:62) A (cid:48) B (cid:48) Φ ≥ , then Φ AB is PPT-inducing. Another sufficient condition can be found by using themap decomposition technique developed in [29]. In fact,Φ AB is PPT-inducing if it can be decomposed into Φ AB = (cid:80) k ( O Ak ⊗ Id B ) ◦ Λ ABk , where the maps Λ ABk are positiveand the operations O Ak ⊗ Id B are PPT-inducing. This factcan be also proven by the rule (4) and Proposition 1.
Proposition 2.
A channel Φ A ⊗ Id B with dim H B ≥ dim H A is PPT-inducing if and only if Ω AA (cid:48) Φ is PPT.Proof. Necessity follows from the fact that the max-imally entangled state | Ψ AB + (cid:105) should become PPTwhen acted upon by Φ A ⊗ Id B . Identification of theproper d A -dimensional subspace of H B and H A (cid:48) im-plies Ω AA (cid:48) Φ is PPT. Sufficiency follows from the tensorproduct form of the Choi operator, namely, Ω ABA (cid:48) B (cid:48) Φ ⊗ Id =Ω AA (cid:48) Φ ⊗ | Ψ BB (cid:48) + (cid:105)(cid:104) Ψ BB (cid:48) + | , which implies Ω ( A ) (cid:62) BA (cid:48) B (cid:48) Φ ⊗ Id =Ω ( A ) (cid:62) A (cid:48) Φ ⊗ | Ψ BB (cid:48) + (cid:105)(cid:104) Ψ BB (cid:48) + | ≥ AA (cid:48) Φ is PPT.Thus, Ω ( A ) (cid:62) BA (cid:48) B (cid:48) Φ ⊗ Id is positive, which guarantees the PPT-inducing behavior of the channel Φ AB by Corollary 1.To some extent, Proposition 2 reproduces the result ofRef. [11] and serves as a sufficient condition for the mapΦ to be entanglement binding. Examples of PPT butentangled Choi states Ω AA (cid:48) Φ are given in Ref. [11].A map Φ AB that maps density operators to undistill-able ones is called distillation-prohibiting. As PPT statescannot be distilled [6], PPT-inducing maps form a convexsubset of distillation-prohibiting ones.The following example illustrates the relation betweenlocal entanglement binding channels and distillation-prohibiting ones. Example 1.
Consider a local channel Φ q ⊗ Φ q actingon two qutrits, where Φ q is depolarizing: Φ q [ X ] = qX +(1 − q )tr[ X ] I , q ≥ − . Suppose | ψ (cid:105) = (cid:80) i =1 √ λ i | ϕ i (cid:105)| χ i (cid:105) is the Schmidt decomposition of an input pure state | ψ (cid:105) ,i.e. (cid:104) ϕ i | ϕ j (cid:105) = (cid:104) χ i | χ j (cid:105) = δ ij , 0 ≤ λ i ≤
1, and (cid:80) i =1 λ i =1. Written in the basis {| ϕ i (cid:105)| χ j (cid:105)} , the density operatorΦ q ⊗ Φ q [ | ψ (cid:105)(cid:104) ψ | ] is a sparse matrix, so one can readily findeigenvalues of its partial transpose. For a fixed q , theminimal possible eigenvalue is achieved if λ = λ = and λ = 0 (up to the permutation of indexes). Exploringpositivity of the minimal eigenvalue, we find that Φ q ⊗ Φ q is PPT-inducing if and only if q ≤ √ √ ≈ . q is known to be entanglement binding if andonly if q ≤ = 0 .
25 [32]. A gap between these two valuesshows that a local channel Φ ⊗ Φ can be PPT-inducing(distillation-prohibiting) even if neither of channels Φ orΦ is entanglement binding. (cid:4) Assuming that the state | ψ (cid:105) = √ ( | ϕ (cid:105)| χ (cid:105) + | ϕ ⊥ (cid:105)| χ ⊥ (cid:105) )of Schmidt rank 2 minimizes the eigenvalues of the partialtranspose of Φ q ⊗ Φ q [ | ψ (cid:105)(cid:104) ψ | ] for a general depolarizingmap Φ q : M d (cid:55)→ M d , Φ q [ X ] = qX + (1 − q )tr[ X ] Id , q ≥ − d − , we readily obtain the following result. Conjecture . Φ q ⊗ Φ q is PPT-inducing if q ≤ √ d +1+ √ .On the other hand, Φ q is entanglement binding if andonly if q ≤ d +1 [32]. The bound on parameter q in theabove Conjecture was first found in Ref. [29] in connec-tion with the search of robust entangled states. VI. CONCLUSIONS
We have introduced the concepts of PPT-inducing anddistillation-prohibiting channels as those that act on acomposite system AB and result in PPT and undistill-able output states, respectively. When a one-sided noisyprocess Φ A ⊗ Id B is distillation-prohibiting, then our defi-nition naturally leads to the notion of entanglement bind-ing channel Φ A . We have characterized PPT-inducingchannels and found necessary and sufficient conditionsfor Φ AB to be PPT-inducing in terms of Choi operators;however, distillation-prohibiting channels still need fur-ther analysis.Also, we have demonstrated that Φ ⊗ Φ can bePPT-inducing (distillation-prohibiting) even if neitherof channels Φ or Φ is entanglement binding. Allthese results show that entanglement binding channelsare analogues of entanglement breaking ones, whereasdistillation-prohibiting channels are analogues of entan-glement annihilating ones: in both cases the notion ofentanglement is merely replaced by the notion of distil-lation capability.Recent progress in the description of the sets of PPTand undistillable states [33, 34] may stimulate a fur-ther characterization of PPT-inducing and distillation-prohibiting channels. Future investigation of continuous-variable systems may follow the same line of reasoning asin Refs. [29, 35] because a necessary and sufficient condi-tion for PPT continuous-variable states is known [36]. Acknowledgments
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