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Convertibility Between Two-Qubit States Using Stochastic Local Quantum Operations Assisted by Classical Communication

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Convertibility Between Two-Qubit States Using Stochastic Local Quantum Operations Assisted by Classical Communication 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.
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