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arXiv:1912.12099 · Lobachevskii Journal of Mathematics

Non-Commutative Graphs in the Fock Space over One-Particle Hilbert Space

Grigori Gennadievich Amosov, Мокеев Александр Сергеевич

Published Apr 1, 2020• 4 citations
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Abstract

Abstract In the present paper we continue our study of non-commutative operator graphs in infinite-dimensional spaces. We consider examples of the non-commutative operator graphs generated by resolutions of identity corresponding to the Heisenberg–Weyl group of operators acting on the Fock space over one-particle state space. The problem of quantum error correction for such graphs is discussed.

DOI
10.1134/s1995080220040034

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Abstract In the present paper we continue our study of non-commutative operator graphs in infinite-dimensional spaces. We consider examples of the non-commutative operator graphs generated by resolutions of identity corresponding to the Heisenberg–Weyl group of operators acting on the Fock space over one-particle state space. The problem of quantum error correction for such graphs is discussed.
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Given a quantum channel it is possible to define the non-commutative operator graph whose properties determine a possibility of error-free transmission of information via this channel. The corresponding graph has a straight definition through Kraus operators determining quantum errors. We are discussing the opposite problem of a proper definition of errors that some graph corresponds to. Taking into account that any graph is generated by some POVM we give a solution to such a problem by means of the Naimark dilatation theorem. Using our approach we construct errors corresponding to the graphs generated by unitary dynamics of bipartite quantum systems. The cases of POVMs on the circle group Zn{\mathbb{Z}}_{n}Zn​ and the additive group R\mathbb{R}R are discussed. As an example we construct the graph corresponding to the errors generated by dynamics of two mode quantum oscillator.