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arXiv:2004.11777

One-Way LOCC Indistinguishable Lattice States via Operator Structures

Comfort Mintah, David W. Kribs, Michael Nathanson, Rajesh Pereira

Published Apr 24, 2020• 0 citations
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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 one-way 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.

DOI
10.48550/arxiv.2004.11777

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We study the finite dimensional C∗C^{*}C∗-algebras and their representation theory. The physical description of quantum local operations and classical communication (LOCC) and its schematics are presented. Focusing on the mathematical description of one-way LOCC, we give detailed analysis of recently derived operator relations in quantum information theory. We also show how functional analytic tools such as operatorx systems, operator algebras, and Hilbert C∗C^{*}C∗-modules all naturally emerge in this setting. We make use of these structures to derive some key results in one-way LOCC. Perfect distinguishability of one-way LOCC versus arbitrary quantum operations is analyzed. It turns out that they are equivalent for several families of operators that appear jointly in matrix and operator theory and quantum information theory. The main results of this work are contained in the paper \\citep{comfort}.
Abstract Most of known locally distinguishable sets of maximally entangled states (MESs) can be perfectly distinguished with one-way local operations and classical communication (1-LOCC), and there are not many known locally distinguishable sets which require two-way LOCC (2-LOCC). Recently some such sets of MESs have been built, that is, they can be locally distinguished only with 2-LOCC. In this work, we show that there exist a lot of locally distinguishable MESs which require 2-LOCC (denote such sets by 2-LOCC sets). Firstly generalized Bell states are employed to construct lots of 1-LOCC indistinguishable sets of MESs in the quantum systems C a + ( a + 1 ) r ⊗ C a + ( a + 1 ) r and C a + ( a + 2 ) r ⊗ C a + ( a + 2 ) r . Then we build a lot of 2-LOCC sets by using Weyl commutation relation and two constructive methods. Especially, it is shown that there are at least 8 and 105 2-LOCC sets in quantum systems C 10 ⊗ C 10 and C 19 ⊗ C 19 respectively.
We bring together in one place some of the main results and applications from our recent work on quantum information theory, in which we have brought techniques from operator theory, operator algebras, and graph theory for the first time to investigate the topic of distinguishability of sets of quantum states in quantum communication, with particular reference to the framework of one-way local quantum operations and classical communication (LOCC). We also derive a new graph-theoretic description of distinguishability in the case of a single-qubit sender.
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In this paper, we mainly consider the local indistinguishability of the set of mutually orthogonal bipartite generalized Bell states (GBSs). We construct small sets of GBSs with cardinality smaller than ddd which are not distinguished by one-way local operations and classical communication (1-LOCC) in d⊗dd\otimes dd⊗d. The constructions, based on linear system and Vandermonde matrix, is simple and effective. The results give a unified upper bound for the minimum cardinality of 1-LOCC indistinguishable set of GBSs, and greatly improve previous results in [Zhang \emph{et al.}, Phys. Rev. A 91, 012329 (2015); Wang \emph{et al.}, Quantum Inf. Process. 15, 1661 (2016)]. The case that ddd is odd of the results also shows that the set of 4 GBSs in 5⊗55\otimes 55⊗5 in [Fan, Phys. Rev. A 75, 014305 (2007)] is indeed a 1-LOCC indistinguishable set which can not be distinguished by Fan's method.