PHYSICAL REVIEW A 77, 012332 ͑2008͒
Convertibility between two-qubit states using stochastic local quantum operations assisted by classical communication
Yeong-Cherng Liang,1,* Lluís Masanes,2,† and Andrew C. Doherty1,‡ 1School of Physical Sciences, University of Queensland, Queensland 4072, Australia 2Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom ͑Received 29 October 2007; published 25 January 2008͒ In this paper we classify the four-qubit states that commute with U U V V, where U and V are arbitrary members of the Pauli group. We characterize the set of separable states for this class, in terms of a finite number of entanglement witnesses. Equivalently, we characterize the two-qubit, Bell-diagonal-preserving, completely positive maps that are separable. These separable completely positive maps correspond to protocols that can be implemented with stochastic local operations assisted by classical communication ͑SLOCC͒. This allows us to derive a complete set of SLOCC monotones for Bell-diagonal states, which, in turn, provides the necessary and sufficient conditions for converting one two-qubit state to another by SLOCC.
DOI: 10.1103/PhysRevA.77.012332 PACS number͑s͒: 03.67.Mn
I. INTRODUCTION now known as stochastic LOCC ͑SLOCC͓͒9͔. In this case, it was shown by Vidal ͓10͔ that in the single copy scenario, a Entanglement has, unmistakeably, played a crucial role in pure state ͉⌿͘ can be locally transformed to ͉⌽͘ with non- many quantum information processing tasks. Despite the zero probability if and only if the Schmidt rank of ͉⌿͘ is various separability criteria that have been developed, deter- higher than or equal to that of ͉⌽͑͘see also Ref. ͓9͔͒. mining whether a general multipartite mixed state is en- The analogous situation for mixed quantum states is not tangled is far from trivial. In fact, computationally, the prob- as well understood even for two-qubit systems. If it were lem of deciding if a quantum state is separable has been possible to obtain a singlet state by SLOCC from a single proven to be nondeterministic polynomial-time hard ͑ ͓͒ ͔ copy of any mixed state, it would be possible to convert any NP-hard 1 . mixed state to any other state ͓11͔. However, as was shown To date, separability of a general bipartite quantum state ͓ ͔͑ ͓ ͔͒ ϫ ϫ ͓ ͔ by Kent et al. 12 see also Ref. 13 , the best that one can is fully characterized only for dimension 2 2 and 2 3 2 . do—in terms of increasing the entanglement of formation For higher dimensional quantum systems, there is no single ͓14͔—is to obtain a Bell-diagonal state with higher but gen- criterion that is both necessary and sufficient for separability. erally nonmaximal entanglement. In fact, apart from some Nevertheless, for quantum states that are invariant under rank deficient states, this conversion process is known to be some group of local unitary operators, separability can often ͑ ͓͒ ͔ ͓ ͔ invertible with some probability 15 . Hence, most two- be determined relatively easily 3–6 . qubit states are known to be SLOCC equivalent to a unique On the other hand, it is often of interest in quantum infor- ͓16͔ Bell-diagonal state of maximal ͓17͔ entanglement mation processing to determine if a given state can be trans- ͓12,15,18͔. formed to some other desired state by local operations. In- ͑ ͒ In this paper, we will complete the picture of two-qubit deed, convertibility between two entangled states using convertibility under SLOCC by providing the necessary and local quantum operations assisted by classical communica- ͑ ͒ sufficient conditions for converting among Bell-diagonal tion LOCC is closely related to the problem of quantifying states. This characterization of the separable completely the entanglement associated to each quantum system. Intu- ͑ ͒ ͑ ͒ positive maps CPM that take Bell-diagonal states to itively, one expects that a single copy entangled state can Bell-diagonal states has other applications. Specifically, it be locally and deterministically transformed to a less was required in the proof of our recent work ͓19͔ which entangled one but not the other way around. ͓ ͔ showed that all bipartite entangled states display a certain This intuition was made concrete in Nielsen’s work 7 kind of hidden nonlocality ͓20͔. ͓We show that a bipartite where he showed that a single copy of a bipartite pure state quantum state violates the Clauser-Horne-Shimony-Holt ͉⌿͘ can be locally and deterministically transformed to an- ͑ ͒ ͓ ͔ ͉⌽͘ ͉⌽͘ CHSH inequality 21 after local preprocessing with some other bipartite state , if and only if takes equal or non-CHSH violating ancilla state if and only if the state is lower values for a set of functions known as entanglement ͔ ͓ ͔ entangled. Thus this paper completes the proof of that monotones 8 . One can, nevertheless, relax the notion of result. convertibility by only requiring that the conversion succeeds The structure of this paper is as follows. In Sec. II, we with some nonzero probability. Such transformations are will start by characterizing the set of separable states com- muting with U U V V, where U and V are arbitrary members of the Pauli group. Then, after reviewing the one- *[email protected] to-one correspondence between separable maps and sepa- †[email protected] rable quantum states in Sec. III A, we will derive, in Sec. ‡[email protected] III B, the full set of Bell-diagonal preserving SLOCC trans-
1050-2947/2008/77͑1͒/012332͑9͒ 012332-1 ©2008 The American Physical Society LIANG, MASANES, AND DOHERTY PHYSICAL REVIEW A 77, 012332 ͑2008͒ formations. A complete set of SLOCC monotones are then derived in Sec. III C to provide the necessary and sufficient conditions for converting a Bell-diagonal state to another. HA ⊗ HB This will then lead us to the necessary and sufficient condi- tions that can be used to determine if a two-qubit state can be HA ⊗ HB converted to another using SLOCC transformations. Finally, we conclude the paper with a summary of results. Throughout, the ͑i, j͒th entry of a matrix W is denoted as ͓ ͔ ͑ ͓͔ ͒ W ij likewise i for the ith component of a vector . More- FIG. 1. A schematic diagram for the subsystems constituting . over, I is the identity matrix and ⌸ is used to denote a Subsystems that are arranged in the same row in the diagram have projector. U U symmetry and hence are represented by Bell-diagonal states ͓5͔͑see text for details͒. In this paper, we are interested in states that are separable between subsystems enclosed in the two dashed II. FOUR-QUBIT SEPARABLE STATES WITH U‹U‹V boxes. ‹V SYMMETRY any operator acting on the same Hilbert space H and hav- Let us begin by reminding the reader of an important ϫ property of two-qubit states which commute with all unitar- ing the same symmetry admits a 4 4 matrix representation ies of the form U U, where U are members of the Pauli M via group. The Pauli group is generated by the Pauli matrices ͕ ͖ 4 4 i i=x,y,z, and has 16 elements. The representation U U de- ͓ ͔ ⌸ ⌸ ͑ ͒ composes onto four one-dimensional irreducible representa- = ͚ ͚ M ij i j, 4 tions, each acting on the subspace spanned by one vector of i=1 j=1 the Bell basis ͓ ͔ where M ij is now not necessarily non-negative. When there 1 is no risk of confusion, we will also refer to r and M, respec- ͉⌽ ͘ϵ ͉͑0͉͘0͘ Ϯ ͉1͉͘1͒͘, ͑1͒ 1 ͱ tively, as a state and an operator having the aforementioned 2 2 symmetry. Evidently, in this representation, an operator is non- 1 ͉⌽ ͘ϵ ͉͑ ͉͘ ͘ Ϯ ͉ ͉͘ ͒͘ ͑ ͒ negative if and only if all entries in the corresponding 4 3 ͱ 0 1 1 0 . 2 4 2 ϫ4 matrix M are non-negative. Notice also that by appro- ⌸ ͓ ͔ priate local unitary transformation, one can swap any i with This implies that 5 any two-qubit state which commutes ⌸ ⌸ with U U can be written as =͚4 ͓r͔ ⌸ , where ⌸ any other j, j i while keeping all the other k, k i, j i=1 i i i unaffected. Here, the term local is used with respect to the A ϵ͉⌽ ͗͘⌽ ͉. With this information in mind, we are now ready i i and B partitioning. Specifically, via the local unitary to discuss the case that is of our interest. transformation We would like to characterize the set of four-qubit states which commute with all unitaries U U V V, where U 1 and V are members of the Pauli group. Let us denote this set ͑ ͒ ͑ ͒ I2 +i z I2 +i z : i =1, j =2, of states by and the state space of ʦ as HӍHAЈ 2 H H H H H BЈ AЉ BЉ, where AЈ, BЈ, etc., are Hilbert 1 ϵ ͑ ͒ ͑ ͒ ͑ ͒ spaces of the constituent qubits. In this notation, both the Vij x + z x + z : i =2, j =3, 5 H H H 2 subsystems associated with AЈ BЈ and that with AЉ Ά · 1 HBЉ have U U symmetry and hence are linear combina- ͑ ͒ ͑ ͒ I2 +i z I2 −i z : i =3, j =4, tions of Bell-diagonal projectors ͓5͔. 2 Our aim in this section is to provide a full characterization H ϵH H ⌸ ⌸ of the set of that are separable between A AЈ AЉ one can swap i and j while leaving all the other Bell and HB ϵHBЈ HBЉ ͑see Fig. 1͒. Throughout this section, a projectors unaffected. In terms of the corresponding 4ϫ4 state is said to be separable if and only if it is separable matrix representation, the effect of such local unitaries on between HA and HB. amounts to permutation of the rows and/or columns of M. The symmetry of allows one to write it as a non- For brevity, in what follows, we will say that two matrices M negative combination of ͑tensored͒ Bell projectors, and MЈ are local-unitarily equivalent if we can obtain M by Ј 4 4 simply permuting the rows and/or columns of M and vice versa. A direct consequence of this observation is that if r ͚ ͚ ͓r͔ ⌸ ⌸ , ͑3͒ = ij i j represents a separable state, so is any other rЈ that is obtained i=1 j=1 from r by independently permuting any of its rows and/or where the Bell projector before and after the tensor product, columns. respectively, acts on HAЈ HBЈ and HAЉ HBЉ ͑Fig. 1͒. Before we state the main result of this section, let us Thus, any state ʦ can be represented in a compact man- introduce one more definition. P ʚ ner, via the corresponding 4ϫ4 matrix r. More generally, Definition 1. Let s be the convex hull of the states
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1000 transposition requirement ͓2͔ for separable states. To complete the proof of Theorem 2, it remains to show 1 0100 D ϵ , that W2, W3, W4 give rise to Hermitian matrices 0 40010 4 4 0001 ͓ ͔ ͑⌸ ⌸ ͒͑͒ Zw,k = ͚ ͚ Wk ij i j 8 i=1 j=1 1100 ͑ ͒Ն that are valid entanglement witnesses, i.e., tr sZw,k 0 for 1 1100 ʦ G ϵ , ͑6͒ any separable s . It turns out that this can be proved with 0 40000 the help of the following lemma from Ref. ͓25͔. 0000 Lemma 3. For a given Hermitian matrix Zw acting on HA HB, with dim͑HA͒=dA and dim͑HB͒=dB, if there ex- + m and the states that are local-unitarily equivalent to these two. ists m,nʦZ , positive semidefinite Z acting on HA Simple calculations show that with respect to the A and B Hn and a subset s of the m+n tensor factors such that ͓ ͔ P B partitioning, D0, G0 are separable 22 . Hence, s is a sepa- ͑ m−1 Z n−1͒ rable subset of . The main result of this section consists of A B IdA w IdB A B showing the converse, and hence the following theorem. T P = A B͑Z s͒A B, ͑9͒ Theorem 2. s is the set of states in that are separable A B with respect to the , partitioning. where A is the projector onto the symmetric subspace of P m T Now, we note that s is a convex polytope. Its boundary HA ͑likewise for B͒ and ͑Z͒ s refers to partial transposi- is therefore described by a finite number of facets ͓23͔. Z tion of with respect to the subsystem s, then Zw is a valid Hence, to prove the above theorem, it suffices to show that H H ͑ ͒Ն entanglement witness across A and B, i.e., tr Zw sep 0 all these facets correspond to valid entanglement witnesses. for any state that is separable with respect to the A and W ͕ ͖ sep Denoting the set of facets by = Wi . Then, using the soft- B partitioning. ware PORTA ͓24͔, the nontrivial facets were found to be A Proof. Denote by k the subsystem associated with the equivalent under local unitaries to one of the following: H Hm B kth copy of A in A ; likewise for l. To prove the above ͉␣͘ʦH ͉͘ʦH ͑ ͒ 111−1 lemma, let A and B be unit vectors, and for definiteness, let s=B then it follows that 111−1 n W ϵ , ͗␣͉͉͗ ͉␣͉͘͘ 1 111−1 Zw ͗␣͉m͉͗n͑ m−1 Z n−1͉͒␣͘m͉͘n −1 −1 −1 1 = IdA w IdB m n T m n = ͗␣͉ ͉͗ ͓A B͑Z s͒A B͔͉␣͘ ͉͘ 110−1 ͗␣͉m͉͗n͑ZTB ͉͒␣͘m͉͘n 001 0 = n W ϵ , ͘ء͉ Z͉␣͘m͉͘n−1͉ء͗ 001 0 = ͗␣͉m͉͗n−1 2 001 0 Ն 0, ͘ is the complex conjugate of ͉͘. We have madeء͉ where 3 31−1 m m use of the identity A͉␣͘ =͉␣͘ ͑likewise for B͒ in the 3−11 3 ͑ ͒ W ϵ , second and third equality, Eq. 9 in the second equality, and 3 1131 the positive semidefiniteness of Z. To cater for general s,we −1 −1 1 −1 just modify the second to last line of the above computation accordingly ͑i.e., to perform complex conjugation on all the states in the set s͒ and the proof will proceed as before. { 331−1 More generally, let us remark that instead of having one Z 3−113 on the right-hand side of Eq. ͑9͒, one can also have a sum of W ϵ . ͑7͒ 4 3−1 1−1 different Z’s, with each of them partially transposed with 11−11 respect to different subsystems s. Clearly, if the given Zw admits such a decomposition, it is also an entanglement wit- Apart from these, there is also a facet W0 whose only non- ness ͓25͔. For our purposes these more complicated decom- ͓ ͔ zero entry is W0 11=1. W0 and the operators local-unitarily positions do not offer any advantage over the simple decom- equivalent to it give rise to positive definite matrices ͓cf. Eq. position given in Eq. ͑9͒. ͑8͔͒, and thus correspond to trivial entanglement witnesses. By solving some appropriate semidefinite programs ͓26͔, B On the other hand, it is also not difficult to verify that W1 we have found that when m=3, n=2, and s= 2, there exist Z Ն ͑ ͒ ͑and operators equivalent under local unitaries͒ are decom- some k 0, such that Eq. 9 holds true for each ʦ͕ ͖ Z posable and therefore demand that remains positive k 1,2,3,4 . The analytic expression for these k’s will semidefinite after partial transposition. These are all the en- not be reproduced here but are made available in ͓27͔. For tanglement witnesses that arise from the positive partial W2, the fact that the corresponding Zw,2 is a witness can even
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A be verified by considering m=2, n=1, and s= 1. In this sents, up to some normalization constant, the most general ͕͉ ͖͘3 case, dA =dB =4. If we label the local basis vectors by i i=0, LOCC possible on a bipartite quantum system. These are the the corresponding Z reads as SLOCC transformations ͓9͔. To make a connection between the set of SLOCC trans- 4 1 Z ͉ ͗͘ ͉ formations and the set of states that we have characterized in 2 = ͚ zi zi , Sec. II, let us also recall the Choi-Jamiołkowski isomor- 2 i=1 phism ͓29͔ between CPM and quantum states: for every ͑not necessarily separable͒ CPM E:HA HB →HA HB ͉z ͘ = ͉01,0͘ − ͉02,3͘ + ͉11,1͘ + ͉13,3͘ + ͉22,1͘ + ͉23,0͘, in in out out 1 there is a unique—again, up to some positive constant ␣—quantum state E corresponding to E, ͉ ͘ ͉ ͘ ͉ ͘ ͉ ͘ ͉ ͘ ͉ ͘ ͉ ͘ z2 = 10,3 + 11,2 + 20,0 + 22,2 − 31,0 + 32,3 , + + + + E = ␣ E I͉͑⌽ ͘A ͗⌽ ͉ ͉⌽ ͘B ͗⌽ ͉͒, ͑12͒ in in ͉ ͘ ͉ ͘ ͉ ͘ ͉ ͘ ͉ ͘ ͉ ͘ ͉ ͘ z3 = 00,0 + 02,2 + 10,1 − 13,2 + 32,1 + 33,0 , + dA where ͉⌽ ͘A ϵ͚ in͉i͘ ͉i͘ is the unnormalized maximally in i=1 + ͉ ͘ ͉ ͘ ͉ ͘ ͉ ͘ ͉ ͘ ͉ ͘ ͉ ͘ entangled state of dimension dA ͑likewise for ͉⌽ ͘B ͒.In z4 = 00,3 + 01,2 − 20,1 + 23,2 + 31,1 + 33,3 , in in Eq. ͑12͒, it is understood that E only acts on one-half of A B + + where we have separated ’s degree of freedom from ’s ͉⌽ ͘A and one-half of ͉⌽ ͘B . Clearly, the state E acts on a degree of freedom with a comma ͓28͔. This completes the in in Hilbert space of dimension dA ϫdA ϫdB ϫdB , where proof for Theorem 2. in out in out dA ϫdB is the dimension of HA HB . out out out out Conversely, given a state E acting on HA HB out out III. SLOCC CONVERTIBILITY OF BELL-DIAGONAL HA HB , the corresponding action of the CPM E on in in STATES some acting on HA HB reads as in in An immediate corollary of the characterization given in 1 T Sec. II is that we now know exactly the set of Bell-diagonal E͑͒ = trA B ͓E͑IA B ͔͒, ͑13͒ preserving transformations that can be performed locally on ␣ in in out out a Bell-diagonal state. In this section, we will make use of the where T denote transposition of in some local bases of Choi-Jamiołkowski isomorphism ͓29͔, i.e., the one-to-one HA HB . For a trace-preserving CPM, it then follows that correspondence between completely positive map ͑CPM͒ in in we must have trA B ͑E͒=␣IA B . A point that should be and quantum state, to make these SLOCC transformations out out in in E ͓ ͑ ͔͒ explicit. This will allow us to derive a complete set of emphasized now is that is a separable map cf. Eq. 10 if ͑ ͒ SLOCC monotones ͓8͔ which, in turn, serve as a set of nec- and only if the corresponding E given by Eq. 12 is sepa- rable across HA HA and HB HB ͓35͔. Moreover, at essary and sufficient conditions for converting one Bell- in out in out diagonal state to another. the risk of repeating ourselves, the map →E͑͒ derived from a separable E can always be implemented locally, al- though it may only succeed with some ͑nonzero͒ probability. A. Separable maps and SLOCC Hence, if we are only interested in transformations that can Now, let us recall some well-established facts about CPM. be performed locally, and not the probability of success in E →E͑͒ ␣ To begin with, a separable CPM, denoted by s takes the mapping , the normalization constant as well as following form ͓30,31͔: the normalization of E becomes irrelevant. This is the con- n vention that we will adopt for the rest of this section. E → ͑ ͒͑ † †͒ ͑ ͒ s: ͚ Ai Bi Ai Bi , 10 i=1 B. Bell-diagonal preserving SLOCC transformations where acts on HA HB , A acts on HA , B acts on HB We shall now apply the isomorphism to the class of states in in i in i in ͓32,33͔. If, moreover, that we have characterized in Sec. II. In particular, if we A A B B A A identify in, out, in, and out with, respectively, Љ, Ј, ͑ ͒†͑ ͒ ͑ ͒ B B ͑ ͒ ͑ ͒ ͚ Ai Bi Ai Bi = I, 11 Љ, and Ј, it follows from Eq. 3 and Eq. 13 that for any i ʦ two-qubit state in, the action of the CPM derived from the map is trace-preserving, i.e., if is normalized, so is the reads as output of the map E ͑͒. Equivalently, the trace-preserving s E → ϰ ͓ ͔ ͑T ⌸ ͒⌸ ͑ ͒ condition demands that the transformation from to E ͑͒ : in out ͚ r ijtr in j i. 14 s i,j can always be achieved with certainty. It is well-known that ͑ ͒ E all LOCC transformations are of the form 10 but the con- Hence, under the action of , any in is transformed to an- verse is not true ͓34͔. other two-qubit state that is diagonal in the Bell basis, i.e., a →E ͑͒ However, if we allow the map s to fail with some Bell-diagonal state. In particular, for a Bell-diagonal in, i.e., Ͻ E ͑͒ probability p 1, the transformation from to s can al- ͓͔ ⌸ ways be implemented probabilistically via LOCC. In other in = ͚ k k, words, if we do not impose Eq. ͑11͒, then Eq. ͑10͒ repre- k
012332-4 CONVERTIBILITY BETWEEN TWO-QUBIT STATES USING … PHYSICAL REVIEW A 77, 012332 ͑2008͒
͓͔ Ն 0, ͚ ͓͔ =1, ͑15͒ 1 −1 k k − σx ⊗ σx − σy ⊗ σy k 0 0 1 the map outputs another Bell-diagonal state −1 −1
E͑ ͒ ϰ ͓͔ ͓ ͔ ⌸ ͑ ͒ −0.5 out = in ͚ j r ij i. 16 i,j − σz ⊗ σz T 0 Φ1 It is worth noting that for general E ʦ,trAЈBЈE is not proportional to the identity matrix, therefore some of the OS 0.5 CPMs derived from ʦ are intrinsically not ͓ ͔ trace-preserving 36 . 1 By considering the convex cone ͓37͔ of separable states P s that we have characterized in Sec. II, we therefore obtain the entire set of Bell-diagonal preserving SLOCC transfor- mations. Among them, we note that the extremal maps, i.e., ͑ ͒ those derived from Eq. ͑6͒, admit simple physical interpreta- FIG. 2. Color online State space for Bell-diagonal states. The T set of physical states is the tetrahedron 0 whose edges are marked tions and implementations. In particular, the extremal sepa- ͑ ͒ rable map for D , and the maps that are related to it by local with thick black lines whereas the set having positive partial trans- 0 pose is another tetrahedron whose edges are marked with thinner unitaries, correspond to permutation of the input Bell projec- ͑ ͒ ⌸ blue lines. The intersection of the two tetrahedra gives rise to the tors i—which can be implemented by performing appropri- octahedron OS ͑blue͒ which is the set of separable Bell-diagonal ate local unitary transformations. The other kind of extremal states. Entangled Bell-diagonal states satisfying Eq. ͑18͒ are con- separable map, derived from G , corresponds to making a 0 tained within the tetrahedron T⌽ ͑green͒, which is discussed further 1 measurement that determines if the initial state is in a sub- in Fig. 3. space spanned by a given pair of Bell states and if successful Ն Ն Ն ͑ ͒ discarding the input state and replacing it by an equal but 1 2 3 4, 18 incoherent mixture of two of the Bell states. This operation can be implemented locally since the equally weighted mix- and determine when it is possible to transform between two ture of two Bell states is a separable state and hence both the such states under SLOCC. ͑ ͒ measurement step and the state preparation step can be Clearly, any entangled Bell-diagonal state can be trans- implemented locally. formed to any separable Bell-diagonal state via SLOCC— one can simply discard the original Bell-diagonal state and prepare the separable state using LOCC. Also separable Bell- C. Complete set of SLOCC monotones for Bell-diagonal states diagonal states can only be transformed among themselves Now, let us make use of the above characterization to with SLOCC. derive a complete set of nonincreasing SLOCC monotones What about transformations among entangled Bell- for Bell-diagonal states. To begin with, we recall that the set diagonal states? To answer this question, we shall adopt the T of normalized Bell-diagonal states forms a tetrahedron 0 in following strategy. First, we will clarify—in relation to Fig. R3, and the set of separable Bell-diagonal states forms an 2—the set of entangled Bell-diagonal states satisfying Eq. O ͑ ͒ T ͓ ͔ ͑ ͒ octahedron S see Fig. 2 that is contained in 0 5 .We 18 . Then, we will make use of the characterization obtained ͓ ͔ ͑ ͗ will follow Ref. 5 and use the expectation values − x in Sec. III B to determine the set of states that can be ob- ͘,−͗ ͘,−͗ ͒͘ as the coordinates of this three- tained from SLOCC transformations when we have an input x y y z z ͑ ͒ ͑ ͒ dimensional space. The coordinates of the four Bell states entangled state satisfying Eq. 18 . After that, we will re- ͕͉⌽ ͖͘4 are then ͑−1,1,−1͒, ͑1,−1,−1͒, ͑−1,−1,1͒, and strict our attention to the subset of these output states satis- i i=1 ͑ ͒ ͑1,1,1͒ respectively. fying Eq. 18 . Once we obtain this, a simple set of necessary Since Bell-diagonal states are convex mixtures of the four and sufficient conditions can be derived to determine if an Bell projectors, we may also label any point in the state entangled Bell-diagonal state can be converted to another. space of Bell-diagonal states by a four-component weight We now take a closer look at the set of entangled Bell- diagonal states, in particular those that satisfy Eq. ͑18͒.In ជ ͑ ͒ vector = 1 , 2 , 2 , 4 such that the corresponding Bell- Fig. 2, the set of entangled Bell-diagonal states is the relative diagonal state reads as ͑ ͒ O complement of the blue octahedron S in the tetrahedron T ͑ ͒ 4 0. In this set, those points that satisfy Eq. 18 are a strict subset contained in the ͑green͒ tetrahedron T⌽ , which has ͑ជ͒ ⌸ ͑ ͒ 1 BD = ͚ i i. 17 ͉⌽ ͘ i the Bell state 1 and the three mixed separable states 1 ͉͑⌽ ͗͘⌽ ͉ ͉⌽ ͗͘⌽ ͉͒ ͑ ͒ Moreover, as remarked above, we can apply local unitary 1i = 1 1 + i i , i = 2,3,4, 19 2 transformation to swap any of the two Bell projectors while leaving others unaffected. Thus, without loss of generality, as its four vertices. In terms of weight vectors, the three we will restrict our attention to Bell-diagonal states such that separable vertices read as
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1 1 1 λ12 0 1 1 1 0 1 0 0 ជ ជ ជ ͑ ͒ − σx ⊗ σx − σy ⊗ σy 12 = , 13 = , 14 = . 20 20 21 20 −0.5 0.5 0 0 1 F 2 1−1 −1 T Ն / ⌽ is the set of Bell-diagonal states satisfying 1 1 2 1 ͑ T ͒ −0.75 which includes both entangled states denoted by E and separable states ͑denoted by F ͒. For the purpose of subse- F − σz ⊗ σz 0 3 F quent discussion, it is important to note that every entangled 1 −0.5 ͑ ͒ T T state satisfying Eq. 18 is in E but not every state in E satisfies Eq. ͑18͒. −0.25 Now, let us consider an entangled Bell-diagonal state ͑ជ͒ 0 BD with weight vector λ14 λ 13 1 ជ 2 = ͑21͒ 3 FIG. 3. ͑Color online͒ Tetrahedron T⌽ ͑with edges marked with ͒ 1 Ն / 4 thick black lines is the set of Bell-diagonal states with 1 1 2. Its ͉⌽ ͑͘ ᭞͒ ͑ ͒ four vertices are the Bell state 1 marked with a and the three satisfying Eq. 18 . Note from the above discussion that ជ ជ ជ ជ separable states , , and . Within T⌽ is the convex polytope ʦT ͑ 12 13 14 1 E. Recall that our goal is to determine the set of en- ជ P, which are points within T⌽ that can be obtained from tangled͒ output states—satisfying Eq. ͑18͒—which can be 1 ជ ͑marked with a Ã͒ by performing SLOCC transformations. Three of obtained from via SLOCC. To achieve that, we will begin P F F F ជ the facets of , namely 1, 2, and 3 are shown with cyan, by first determining the set of output weight vectors ͕Ј͖ green, and light purple colors, respectively; the other facets of P which are in the superset T⌽ . are shown with blue color. 1 In particular, we note that under extremal SLOCC trans- of the polytope ͓23͔. Therefore, the above task can be done, formations associated with G0, and the operators ជ ជЈ local-unitarily equivalent to it ͓cf. Eq. ͑6͒ and Sec. III B͔, for example, by checking if satisfies all the linear equali- ͕ ͖4 ties defining the polytope P. can be brought into any of the separable states 1i i=2 ͓cf. Eqs. ͑19͒ and ͑20͔͒. Similarly, under extremal SLOCC Our real interest, however, is in the set of entangled Bell-diagonal states satisfying Eq. ͑18͒. With some thought, transformations associated with D0, and the operators local- ជ it should be clear that this simplifies the problem at hand unitarily equivalent to it, can be brought into any of the ជ so that we will only need to check that Ј satisfies all following entangled Bell-diagonal states by permuting the ជ weights associated with some of the Bell projectors, of the inequalities ͑facets͒ that contain . From Fig. 3, P ជ it can be seen that only three facets of contain . 1 1 1 F ͕ជ ជ ជ ជ ជ ជ ͖ F These are 1 =conv , ͑34͒ , ͑324͒ , ͑24͒ , ͑234͒ , ͑23͒ , 2 ជ 2 ជ 3 ជ 4 ͑ ͒ = , ͑ ͒ = , ͑ ͒ = , ͕ជ ជ ជ ͖ F ͕ជ ជ ជ ជ ͖ 34 324 24 =conv , 12, ͑34͒ , and 3 =conv , 12, 13, ͑23͒ , where 4 4 3 conv͕ ͖ represents the convex hull formed by the set of ¯ 3 2 2 ͕ ͖͓ ͔ points in ¯ 23 . ជ Recall that each vector ͑ ͒ listed in Eq. ͑22͒ is obtained ¯ 1 1 ͑ ͒ by performing the appropriate permutation ¯ on all but the ជ 4 ជ 3 ជ ͑ ͒ = , ͑ ͒ = . ͑22͒ F 234 23 first component of . Hence 1 is a facet of constant 1. 2 2 After some simple algebra, the inequalities associated with F ͓ ͔ F ͓ ͔ 3 4 2 39 and 3 40 can be shown to be, respectively, Evidently, any convex combinations of the vectors listed in + F : 3 4 ͑͗ ͘ − ͗ ͒͘ + ͗ ͘ Յ 1, ͑ ͒ ͑ ͒ ជ ͑ 2 x x y y z z Eqs. 20 – 22 are also attainable from using nonextre- 1 − 2 ͒ T mal SLOCC. Moreover, within ⌽ , only convex combina- ͑ ͒ 1 ជ 23 tions of these states, denoted by P, are attainable from using SLOCC. P is thus a convex polytope with vertices 1−2 +2 F :͗ ͘ + ͗ ͘ − 1 4 ͗ ͘ Յ 1. given by the union of vectors listed in Eqs. ͑20͒–͑22͒. 3 x x z z y y ជ 1−2 2 −2 3 Then, to determine if can be transformed to another ជ ជ ͑24͒ ЈʦT⌽ amounts to deciding if ЈʦP. It is a well-known 1 ជ fact that a convex polytope can also be described by a finite Imposing the requirement that Ј satisfies these inequalities set of inequalities that are associated with each of the facets gives, respectively,
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Ј 1−2 2 1−2 where ND is expressed in the standard product basis and b Ն 2 1 Յ is unique. + Ј + Ј 2 3 4 3 4 Moreover, as was shown in Ref. ͓15͔, the unique Bell- and diagonal state in case ͑1͒ is the state with maximal entangle- ment that can be obtained from the original two-qubit state Ј Ј 1−2 2 −2 3 1−2 −2 ͑ ͒ Ն 2 3 . using SLOCC. The two-qubit state associated with case 2 is Ј 4 4 clearly a separable one since a separable state is, and can only be, SLOCC equivalent to another separable state. Together with the requirement imposed by F , we see that by 1 The situation for case ͑3͒ is somewhat more complicated defining and the original two-qubit states associated with this case are ជ either of rank 3 or 2 ͑in the case of b=1/2͓͒13,15,18͔.By E ͑͒ϵ , ͑25͒ 1 1 very inefficient SLOCC transformations—quasidistillation ͓38͔—the entanglement in the equivalent state can be ND ជ 1−2 2 E ͑͒ϵ , ͑26͒ maximized by converting it into the following Bell-diagonal 2 3 + 4 state:
00 00 ជ 1−2 2 −2 3 E ͑͒ϵ , ͑27͒ 3 1 01−2b 0 4 = . ͑32͒ ND 20 −2b 1 0 the intercovertibility between two entangled Bell-diagonal states can be succinctly summarized in the following 00 00 theorem. However, it remains unclear from existing results Theorem 4. Let and Ј be two entangled Bell-diagonal ͓13,15,18,38͔ if this process is reversible ͓41͔. In this regard, ជ ជ states with, respectively, weight vectors and Ј satisfying we have found that the reverse process can indeed be carried Eq. ͑18͒. Transformation from to Ј via SLOCC is possible out via a separable map with two terms involved in the Kraus if and only if decomposition. In particular, a possible form of the Kraus operators associated with this separable map reads ͓Eq. ͑10͔͒ ͑ជ͒ Ն ͑ជ ͒ ͑ ͒ E1 E1 Ј , 28 as ͱ 2 / / ជ ជ ͩ−2b + 1+4b −12 ͪ ͩ112 ͪ E ͑͒ Ն E ͑Ј͒, ͑29͒ A1 = , B1 = , 2 2 10 10 ជ ជ ͑͒ Ն ͑Ј͒ ͑ ͒ ͱ 2 E3 E3 . 30 ͩ2b − 1+4b 1/2 ͪ ͩ 11/2 ͪ A2 = , B2 = . ͕ ͑ជ͖͒3 10 −1 0 In other words, Ei i=1 is a complete set of SLOCC mono- tones for entangled Bell-diagonal states satisfying Eq. ͑18͒. Thus, a two-qubit state that is SLOCC equivalent to ND is also SLOCC equivalent to a unique Bell-diagonal state ND. IV. SLOCC CONVERTIBILITY OF TWO-QUBIT By further local unitary transformation, we can bring ND STATES into a form that satisfies Eq. ͑18͒. Hence, this leads us to the With Theorem 4, it is just another small step to determine following theorem. Ј Theorem 7. All entangled two-qubit states are SLOCC if a two-qubit state can be converted to another, say ͑ ͒ using SLOCC. To this end, let us first recall the following equivalent to a unique Bell-diagonal state satisfying Eq. 18 . ͓ ͔ With this theorem, one can now readily determine if an definition from Ref. 9 . Definition 5. Two states and Ј are said to be SLOCC entangled two-qubit state can be converted to another, say, ͑ជ͒ ͑ជ ͒ equivalent if can be converted to Ј via SLOCC with non- Ј, using SLOCC. For that matter, let BD and BD Ј be, zero probability and vice versa. respectively, the unique Bell-diagonal state satisfying Eq. Next, we recall the following theorem, which can be de- ͑18͒ that is SLOCC equivalent to and Ј. Then, it follows duced from Theorem 1 in Ref. ͓18͔͑see also Theorems 1–3 from Theorem 4 that can be transformed to Ј using in Ref. ͓15͔͒. SLOCC if and only if the corresponding weight vectors of ជ ជ Theorem 6. A two-qubit state is SLOCC equivalent to the associated Bell-diagonal states and Ј satisfy Eqs. either ͑1͒ a unique Bell-diagonal state satisfying Eq. ͑18͒, ͑2͒ ͑28͒–͑30͒. In other words, the SLOCC convertibility of two a separable state, or ͑3͒ a ͑normalized͒ non-Bell-diagonal two-qubit states can be decided via the following theorem. state of the form ͑ជ͒ ͑ជ ͒ Theorem 8. Let BD and BD Ј be, respectively, the ͑ ͒ 20 00 Bell-diagonal state satisfying Eq. 18 that is SLOCC equiva- lent to and Ј. can be locally transformed onto Ј with 1 012b 0 ͑ ͒ nonzero probability if and only if ͑1͒ Ј is separable or ͑2͒ ND = , 31 40 2b 1 0 ជ ជЈ is entangled and the associated weight vectors and sat- 00 00 isfy Eqs. ͑28͒–͑30͒.
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Schematically, if neither nor Ј are separable and if Eqs. Equivalently, this characterization has also given us the ͑28͒–͑30͒ are satisfied, then one possible way of transform- complete set of separable, Bell-diagonal preserving, com- ing to Ј via SLOCC is by performing the following chain pletely positive maps. This has enabled us to derive a com- of conversions: plete set of SLOCC monotones for Bell-diagonal states, which can be used to determine if a Bell-diagonal state can → ͑ជ͒ → ͑ជ ͒ → BD BD Ј Ј, be converted to another using SLOCC. We have then supplemented the result on SLOCC equiva- whereas if any one of Eqs. ͑28͒–͑30͒ is not satisfied, then lence presented in Refs. ͓15,18͔ to arrive at the conclusion → ” Ј. that all entangled two-qubit states are SLOCC equivalent to a unique Bell-diagonal state. Combining this with the SLOCC monotones that we have derived immediately leads us to V. DISCUSSION AND CONCLUSION some simple necessary and sufficient criteria on the SLOCC convertibility between two-qubit states. In this paper, we have investigated the biseparability of the set of four-qubit states commuting with U U V V ACKNOWLEDGMENTS where U and V are arbitrary members of the Pauli group. These are essentially convex combination of two ͑not neces- The authors would like to thank Guifré Vidal and Frank sarily identical͒ copies of Bell states. Evidently, these states Verstraete for helpful discussions. One of the authors are all separable across the two copies. For the other bipar- ͑Y.-C.L.͒ would also like to thank Chris Foster and Paulo titioning, we have found that the separable subset is a convex Mendonça for their help with the figures. This work is sup- polytope and hence can be described by a finite set of en- ported by the EU Project QAP ͑Contract No. IST-3-015848͒ tanglement witnesses. and the Australian Research Council.
͓1͔ L. Gurvits, Proceedings of the Thirty-Fifth ACM Symposium ͓20͔ S. Popescu, Phys. Rev. Lett. 74, 2619 ͑1995͒; N. Gisin, Phys. on Theory of Computing ͑ACM, New York, 2003͒, pp. 10–19; Lett. A 210, 151 ͑1996͒. L. M. Ioannou, Quantum Inf. Comput. 7, 335 ͑2007͒. ͓21͔ J. F. Clauser, M. A. Horne, A. Shimony, and R. Holt, Phys. ͓2͔ A. Peres, Phys. Rev. Lett. 77, 1413 ͑1996͒; M. Horodecki, P. Rev. Lett. 23, 880 ͑1969͒; J. S. Bell, Foundation of Quantum Horodecki, and R. Horodecki, Phys. Lett. A 223,1͑1996͒. Mechanics. Proceedings of the International School of Physics ͓3͔ R. F. Werner, Phys. Rev. A 40, 4277 ͑1989͒. ‘Enrico Fermi’, Course IL ͑Academic, New York, 1971͒, pp. ͓4͔ M. Horodecki and P. Horodecki, Phys. Rev. A 59, 4206 171–181. ͑1999͒. ͓22͔ Their separability can be veirfied, for example, by writing ͓5͔ K. G. H. Vollbrecht and R. F. Werner, Phys. Rev. A 64, these operators in full in the product basis via Eq. ͑4͒ and 062307 ͑2001͒. showing that they admit convex decomposition in terms of ͓6͔ T. Eggeling and R. F. Werner, Phys. Rev. A 63, 042111 ͑2001͒. separable states. Alternatively, via the Choi-Jamiołkowski iso- ͓7͔ M. A. Nielsen, Phys. Rev. Lett. 83, 436 ͑1999͒. morphism that will be discussed later in Sec. III A and the ͓8͔ G. Vidal, J. Mod. Opt. 47, 355 ͑2000͒. remarks made towards the end of Sec. III B, one can also see ͓ ͔ 9 W. Dür, G. Vidal, and J. I. Cirac, Phys. Rev. A 62, 062314 that these matrices correspond to separable states. ͑ ͒ 2000 . ͓23͔ B. Grünbaum, Convex Polytopes ͑Springer, New York, 2003͒. ͓ ͔ ͑ ͒ 10 G. Vidal, Phys. Rev. Lett. 83, 1046 1999 . ͓24͔ This software package, which stands for POlyhedron Repre- ͓11͔ C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schuma- sentation Transformation Algorithm, is available at http:// cher, Phys. Rev. A 53, 2046 ͑1996͒. www.zib.de/Optimization/Software/Porta/ ͓12͔ A. Kent, N. Linden, and S. Massar, Phys. Rev. Lett. 83, 2656 ͓25͔ A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri, Phys. Rev. ͑1999͒. Lett. 88, 187904 ͑2002͒; A. C. Doherty, P. A. Parrilo, and F. ͓13͔ L.-X. Cen, F.-L. Li, and S.-Y. Zhu, Phys. Lett. A 275, 368 69 ͑ ͒ ͑2000͒; L.-X. Cen, N.-J. Wu, F.-H. Yang, and J.-H. An, Phys. M. Spedalieri, Phys. Rev. A , 022308 2004 . ͓ ͔ ͑ ͒ Rev. A 65, 052318 ͑2002͒. 26 L. Vandenberghe and S. Boyd, SIAM Rev. 38,49 1996 ;S. ͑ ͓14͔ S. Hill and W. K. Wootters, Phys. Rev. Lett. 78, 5022 ͑1997͒. Boyd and L. Vandenberghe, Convex Optimization Cambridge, ͒ ͓15͔ F. Verstraete, J. Dehaene, and B. DeMoor, Phys. Rev. A 64, New York, 2004 . 010101͑R͒͑2001͒. ͓27͔ http://www.physics.uq.edu.au/qisci/yerng/ ͓ ͔ Z ͑ ͒ ͓16͔ Obviously, unique up to local unitary transformations. 28 Note that to verify 2 against Eq. 9 , one should also rewrite ͑ ͒ ͓17͔ Maximal, in the sense that no SLOCC transformations can Zw,2 obtained in Eq. 8 in the appropriate tensor-product basis H H H H bring the state to another one with higher entanglement of such that Zw,2 acts on AЈ AЉ BЈ BЉ. formation. ͓29͔ A. Jamiołkowski, Rep. Math. Phys. 3, 275 ͑1972͒;M.D. ͓18͔ F. Verstraete, J. Dehaene, and B. DeMoor, Phys. Rev. A 65, Choi, Linear Algebr. Appl. 10, 285 ͑1975͒; V. P. Belavkin and 032308 ͑2002͒. P. Staszewski, Rep. Math. Phys. 24,49͑1986͒. ͓19͔ Ll. Masanes, Y.-C. Liang, and A. C. Doherty, e-print ͓30͔ E. M. Rains, e-print arXiv:quant-ph/9707002. arXiv:quant-ph/0703268. ͓31͔ V. Vedral and M. B. Plenio, Phys. Rev. A 57, 1619 ͑1998͒.
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͓32͔ Following Kraus’ work on CPM ͓33͔, this specific form of the quent discussion, we might as well consider the cone generated
CPM is also known as a Kraus decomposition of the CPM, by Ps. with each Ai Bi in the sum conventionally called the Kraus ͓38͔ M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. A operator associated with the CPM. 60, 1888 ͑1999͒. ͓ ͔ ͑ 33 K. Kraus, State, Effects, and Operations Springer, Berlin, ͓39͔ Strictly, the inequality ͑23͒ is only valid when . When 1983͒; K. Kraus, Ann. Phys. ͑N.Y.͒ 64,311͑1971͒. 3 4 = , F degenerates into an edge of the polytope P.If ͓34͔ C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. 3 4 2 F ជ ជ ជ Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters, Phys. 1 = 2, 2 collapses into a single point = 12= ͑34͒. ͓40͔ T ជ ជ ជ ជ Rev. A 59, 1070 ͑1999͒. Within E, 1−2 2 −2 3 =0 only when = 12 and ͑23͒ = 13.In ជ ជ ͓35͔ J. I. Cirac, W. Dür, B. Kraus, and M. Lewenstein, Phys. Rev. this case, F degenerates into the line joining and . ͑ ͒ 3 12 13 Lett. 86, 544 2001 . ͓41͔ By this, of course, we are referring only to the states given in ͓ ͔ ͑ ͒ 36 The E derived from G0 in Eq. 6 is an example of this sort. In Eq. ͑31͒ that are originally of rank 3, and which becomes rank fact, in this case, if the input state has no support on ⌸ nor 1 2 upon quasidistillation. The states given in Eq. ͑31͒ that are of ⌸ , the map always outputs the zero matrix. 2 rank 2 get quasidistilled to the singlet state and so the process ͓37͔ Since the mapping from any ʦP to a separable CPM via Eq. s is clearly reversible in this case. ͑13͒ is only defined up to a positive constant, for the subse-
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