Local distinguishability of any three quantum states

We prove that any three linearly independent pure quantum states can always be locally distinguished with nonzero probability regardless of their dimension, entanglement or multipartite structure. Almost always, all three states can be unambiguously identified. The only exceptional case, where one state is locally knowable but the other two are not, is found among multi-qubit states. PACS numbers: 89.70.+c, 03.65.−w Global operations on a quantum system can process information in ways that local operations on the system’s parts cannot. All uses of entanglement in quantum information theory flow from this one fact, from teleportation [1] to Shor’s factoring algorithm [2]. However a fundamental question remains unanswered. When is global information about a quantum system also available locally? This question can be formally posed as a local state discrimination task. Given one copy of a system in one of a known set of quantum states {|ψi〉}, how much ‘which state’ information can be gleaned by local operations and classical communication (LOCC), and how much more information is revealed by global measurements? This problem has attracted much attention in recent years, after surprising results showed perfect local distinguishability was not directly linked to entanglement. Bennett and coworkers presented sets of orthogonal unentangled states that were not perfectly locally distinguishable [3]. JW, Short, Hardy and Vedral proved orthogonal pairs of states are always perfectly locally distinguishable, irrespective of their entanglement [4]. There are two natural approaches to quantum state discrimination. Optimal discrimination seeks the best possible guess as to the state of the system [5]. Conclusive discrimination (also called unambiguous discrimination) seeks certain knowledge of the state of the system, balanced against a possibility of failure [6]. In particular, N linearly independent quantum 4 Present address: Department of Physics, Bose Institute, Kolkata 700009, India. E-mail: [email protected] 1751-8113/09/072002+07$30.00 © 2009 IOP Publishing Ltd Printed in the UK 1 J. Phys. A: Math. Theor. 42 (2009) 072002 Fast Track Communication states can be conclusively distinguished by a (N + 1)-outcome generalized measurement, where N outcomes correspond to correct (that is, error free) identification of the quantum states, and the remaining outcome corresponds to an inconclusive result, for which one fails to correctly identify the unknown quantum state. Note that conclusive identification of every state is not possible if the states are linearly dependent [7]. It follows directly from the results of Walgate et al [4] and Virmani et al [8] that local parties can always gain some amount of ‘which state’ information about the pure state of a shared system, and use it to improve their guesswork. Optimal state discrimination is always locally feasible in this sense, although the local optimum may be significantly worse than the global. Conclusive discrimination is more interesting. All pairs of pure quantum states can be conclusively discriminated equally well locally and globally [9]. Generically, a small number of pure states (proportional to the dimension of the subsystems) can be conclusively discriminated with non-negligible probability [15]. But in every multipartite dimensionality there are sets of four pure quantum states that are not conclusively locally distinguishable at all. In this case local parties can never gain certain knowledge of which state they possess; the Bell states are the simplest example of such a set [10]. So two states are always conclusively distinguishable, and four states can be conclusively indistinguishable. We complete the picture by showing that provided they are linearly independent (only linearly independent states are globally distinguishable) three pure quantum states can be conclusively locally distinguished. Local protocols may not succeed as often as global measurements, but they can succeed some of the time. No triplet of pure states, no matter how entangled, conceals any fraction of its ‘which state’ information from local parties with certainty. We present our results in the following framework. A multipartite quantum system Q is shared between n different local parties, each with access to one of n local Hilbert spaces: HQ = ⊗n j=1 Hj . It has been prepared in one of a known set of possible pure states S = {|ψi〉}, each with some nonzero (but potentially unknown) probability pi . The local parties are set the task of discovering with certainty which of the states S they have been given, using only LOCC. We will use the following definitions. Definition 1. A state |ψi〉 ∈ S is conclusively locally identifiable if and only if there is a LOCC protocol whereby with some nonzero probability p > 0 it can be determined that Q was certainly prepared in state |ψi〉. Definition 2. A set of states S is conclusively locally distinguishable if and only if every state in S is conclusively locally identifiable. Conclusive state identification has qualitative links to entanglement. It was proved by Horodecki et al that the states of a complete orthonormal basis are conclusively locally identifiable states if and only if they are product states [11]. We show below in corollary 1 that if no members of an incomplete basis of orthogonal states are conclusively identifiable then the set must be completely entangled. We begin by establishing a necessary and sufficient condition for a set of states to be conclusively locally distinguishable, first proved by Chefles [12]. We outline a simplified version of Chefles’ specific to the case of pure states. We will then show how this condition holds for sets of three states. Lemma 1 (Chefles). Let a multipartite quantum system Q be prepared in one of a set of pure, linearly independent multipartite quantum states S = {|ψi〉}. Let |ψx〉 ∈ S. If and only if there exists a product state |φ〉 such that ∀ i = x〈ψi |φ〉 = 0, and 〈ψx |φ〉 = 0, then |ψx〉 is conclusively locally identifiable in S.