ua nt - p h / 02 08 05 9 v 1 9 A ug 2 00 2 Criterion for local distinguishability of arbitrary bipartite orthogonal states Ping -

In this paper we present a necessary and sufficient condition of distinguishability of bipartite quantum states. It is shown that the operators to reliably distinguish states need only rounds of projective measurements and classical comunication. We also present a necessary condition of distinguishability of bipartite quantum states which is simple and general. With this condition one can get many cases of indistinguishability. The conclusions may be useful in calculating the distillable entanglement and the bound of distillable entanglement. PACS number(s): 03.67.-a, 03.65.ud Typeset using REVTEX E-mail: [email protected] 1 One of the interesting features of non-locality in quantum mechanics is that a set of orthogonal quantum states cannot be distinguished if only a single copy of these states is provided and only local operations and classical communication (LOCC) are allowed, in general. The procedure of distinguishing quantum states locally is: Alice and Bob hold a part of a quantum system, which occupies one of m possible orthogonal states |Ψ1〉 , |Ψ2〉 , ..., |Ψi〉 , ..., |Ψm〉. Alice and Bob know the precise form of these states, but don’t know which of these possible states they actually hold. To distinguish these possible states they will perform some operations locally: Alice first measures her part. Then she tell the Bob her measurement result, according to which Bob measure his part. With the measurement results they can exclude some possibilities of the system [1]. Briefly speaking, the procedure of distinguishing quantum states locally is to exclude all or some possibilities by measurement on the system. Many authors have considered some schemes for distinguishing locally between a set of quantum states [1–7], both inseparable and separable. Bennett et al showed that some orthogonal product states cannot be distinguished by LOCC [2]. Walgate et al showed that any two states can be distinguished by LOCC [1]. For two-qubit systems (or 2⊗ 2 systems), any three of the four Bell states: |A1〉 = 1 √ 2 (|00〉+ |11〉) (1) |A2〉 = 1 √ 2 (|00〉 − |11〉) |A3〉 = 1 √ 2 (|01〉+ |10〉) |A4〉 = 1 √ 2 (|01〉 − |10〉) cannot be distinguished by LOCC if only a single copy is provided [4]. The distinguishability of quantum states has some close connections [8] with distillable entanglement [9] and the information transformation [10]. On one hand, using the upper bound of distillable entanglement, relative entropy entanglement [11] and logarithmic negativity [12], the authors in Ref [4] proved that some states are indistinguishable. On the other hand, using the rules on distinguishability one can discuss the distillable entanglement [8]. So the further analysis for distinguishability is meaningful. In this paper, we will give a necessary and sufficient condition of distinguishability of bipartite quantum states. It is shown that the operators to reliably distinguish states need only rounds of projective measurements and classical comunication. We also present a necessary condition of distinguishability of bipartite quantum states which is simple and general. With this condition one can get many cases of indistinguishability. Consider m possible orthogonal states shared between Alice and Bob. Any protocol to distinguish the m possible orthogonal states can be conceived as successive rounds of measurement and communication by Alice and Bob. Let us suppose Alice is the first person to perform a measurement (Alice goes first [3]), and the first round measurement by Alice can be represented by operators {