One-way LOCC indistinguishable lattice states via operator structures
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One-way LOCC indistinguishable lattice states via operator structures David W. Kribs1,2 · Comfort Mintah1
· Michael Nathanson3 · Rajesh Pereira1
Received: 26 August 2019 / Accepted: 27 April 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Lattice states are a class of quantum states that naturally generalize the fundamental set of Bell states. We apply recent results from quantum error correction and from oneway local operations and classical communication (LOCC) theory that are built on the structure theory of operator systems and operator algebras, to develop a technique for the construction of relatively small sets of lattice states not distinguishable by one-way LOCC schemes. We also present examples, show the construction extends to generalized Pauli states, and compare the construction to other recent work. Keywords Quantum state discrimination · Bell states · Lattice states · Quantum entanglement · Local operations and classical communication · Operator system · Operator algebra · Separating vector Mathematics Subject Classification 47L90 · 46B28 · 81P15 · 81P45 · 81R15
1 Introduction A basic problem in quantum information theory is that of identifying a state from a set of known states on a composite quantum system, utilizing only quantum operations local to the individual subsystems [2,3,9,10]. Many problems in the subject can be seen as special cases of the so-called local operations and classical communication (LOCC) framework, such as quantum teleportation and data hiding [1,7,17]. The restricted problem of quantum state discrimination with only one-way LOCC operations, in
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Comfort Mintah [email protected]
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Department of Mathematics & Statistics, University of Guelph, Guelph, ON N1G 2W1, Canada
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Institute for Quantum Computing and Department of Physics & Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada
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Department of Mathematics and Computer Science, Saint Mary’s College of California, Moraga, CA 94556, USA 0123456789().: V,-vol
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which local operations are performed sequentially on the different subsystems, has been identified as a subproblem of central importance, with special emphasis placed on identifying small sets of indistinguishable states under the paradigm [5,8,13,14,19,21]. An important class of quantum states, called lattice states, is a natural generalization of the fundamental Bell states and has been studied previously in the context of LOCC and positive partial transpose (PPT) measurements; for instance in [4,5,21]. In this paper, we apply recently established results from one-way LOCC theory and quantum error correction [11,12], that are based on a structural analysis of certain operator systems and operator algebras which arise in the LOCC framework, to the state discrimination problem for lattice states. Specifically, we develop a technique for the construction of relatively small sets of lattice states that are indistinguishable under one-way LOCC schemes. We also show how t
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