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Limitations on separable measurements by convex optimization
 Information Theory, IEEE Transactions on
, 2015
"... The paradigm of local operations and classical communication, or LOCC for short, is fundamental within the theory of quantum information. A protocol involving two or more individuals is said to be an LOCC protocol when it may be implemented by means of classical communication among the individuals, ..."
Abstract

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The paradigm of local operations and classical communication, or LOCC for short, is fundamental within the theory of quantum information. A protocol involving two or more individuals is said to be an LOCC protocol when it may be implemented by means of classical communication among the individuals, along with arbitrary quantum operations performed locally. Within the LOCC paradigm, the problem of discrimination of quantum states (bipartite or multipartite) has been extensively studied. In the most typically considered variant of this problem, the two parties are given a single copy of a quantum bipartite state chosen with some probability from a known collection of states and their goal is to identify which state was given by means of an LOCC measurement. Many examples are known of specific choices of pure, orthogonal states for which a perfect discrimination is not possible through LOCC measurements. Some of these examples, along with other general results concerning this problem, may be found in [BW09, BDF + 99, DFXY09, Fan04, GKR + 01, GKRS04, HMM + 06, HSSH03, Nat05, WH02, WSHV00, Wat05, YDY12, YDY14]. As perhaps the simplest example of an instance of this problem where a perfect LOCC discrimination is not possible, one has that the four standard Bell states cannot be perfectly discriminated by LOCC measurements [GKR + 01]. In particular, if the states are selected with uniform probability, it holds that the maximum probability of distinguishing them via LOCC is 1/2. Among the other known examples of collections of orthogonal pure states that cannot be perfectly discriminated by LOCC protocols, the socalled domino state example of [BDF + 99] is noteworthy. The particular relevance of this example lies in the fact that all of these states are product states, demonstrating that entanglement is not a requisite for a set of orthogonal pure states to fail to be perfectly discriminated by any LOCC measurement. The set of measurements that can be implemented through LOCC has an apparently complex mathematical structureno tractable characterization of this set is known, representing a clear obstacle to a better understanding of the limitations of LOCC measurements. For this reason, the state discrimination problem described above is sometimes considered for more tractable classes of measurements that approximate, in some sense, the LOCC measurements. The classes of positivepartialtranspose (PPT) and separable measurements represent two commonly studied