Asymptotic State Discrimination and a Strict Hierarchy in Distinguishability Norms

In this paper, we consider the problem of discriminating quantum states by local operations and classical communication (LOCC) when an arbitrarily small amount of error is permitted. This paradigm is known as asymptotic state discrimination, and we derive necessary conditions for when two multipartite states of any size can be discriminated perfectly by asymptotic LOCC. We use this new criterion to prove a gap in the LOCC and separable distinguishability norms. We then turn to the operational advantage of using two-way classical communication over one-way communication in LOCC processing. With a simple two-qubit product state ensemble, we demonstrate a strict majorization of the two-way LOCC norm over the one-way norm.Comment: Corrected errors from the previous draft. Close to publication for

Round Complexity in the Local Transformations of Quantum and Classical States

A natural operational paradigm for distributed quantum and classical information processing involves local operations coordinated by multiple rounds of public communication. In this paper we consider the minimum number of communication rounds needed to perform the locality-constrained task of entanglement transformation and the analogous classical task of secrecy manipulation. Specifically we address whether bipartite mixed entanglement can always be converted into pure entanglement or whether unsecure classical correlations can always be transformed into secret shared randomness using local operations and a bounded number of communication exchanges. Our main contribution in this paper is an explicit construction of quantum and classical state transformations which, for any given $r$, can be achieved using $r$ rounds of classical communication exchanges but no fewer. Our results reveal that highly complex communication protocols are indeed necessary to fully harness the information-theoretic resources contained in general quantum and classical states. The major technical contribution of this manuscript lies in proving lower bounds for the required number of communication exchanges using the notion of common information and various lemmas built upon it. We propose a classical analog to the Schmidt rank of a bipartite quantum state which we call the secrecy rank, and we show that it is a monotone under stochastic local classical operations.Comment: Submitted to QIP 2017. Proof strategies have been streamlined and differ from the submitted versio

Optimal Entanglement Transformations Among N-qubit W-Class States

We investigate the physically allowed probabilities for transforming one N-partite W-class state to another by means of local operations assisted with classical communication (LOCC). Recently, Kintas and Turgut have obtained an upper bound for the maximum probability of transforming two such states [arXiv:1003.2118v1]. Here, we provide a simple sufficient and necessary condition for when this upper bound can be satisfied and thus when optimality of state transformation can be achieved. Our discussion involves obtaining lower bounds for the transformation of arbitrary W-class states and showing precisely when this bound saturates the bound of [arXiv:1003.2118v1]. Finally, we consider the question of transforming symmetric W-class states and find that in general, the optimal one-shot procedure for converting two symmetric states requires a non-symmetric filter by all the parties