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Proceedings of the Steklov Institute of Mathematics

On Noncommutative Operator Graphs Generated by Resolutions of Identity

Grigori Gennadievich Amosov, Мокеев Александр Сергеевич

Published Jul 1, 2021• 0 citations
Publisher

Abstract

Abstract Noncommutative operator graphs play an important role in the theory of quantum error correction. In this paper, we briefly review recent results devoted to the graphs generated by resolutions of identity for which there exists a quantum error-correcting code. We discuss examples of such graphs and touch upon the problem of describing quantum noise within this theory.

DOI
10.1134/s0081543821020024

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Abstract Noncommutative operator graphs play an important role in the theory of quantum error correction. In this paper, we briefly review recent results devoted to the graphs generated by resolutions of identity for which there exists a quantum error-correcting code. We discuss examples of such graphs and touch upon the problem of describing quantum noise within this theory.
Quantum error correction plays a key role for quantum information transmission and quantum computing. In this work, we develop and apply the theory of noncommutative operator graphs to study error correction in the case of a finite-dimensional quantum system coupled to an infinite-dimensional system. We consider as an explicit example a qubit coupled via the Jaynes-Cummings (JC) Hamiltonian with a bosonic coherent field. We extend the theory of noncommutative graphs to this situation and construct, using Gazeau-Klauder coherent states, the corresponding noncommutative graph. As the result, we find the quantum anticlique, which is the projector on the error-correcting subspace, and analyze it as a function of the frequencies of the qubit and the bosonic field. The general treatment is also applied to the analysis of the error-correcting subspace for certain experimental values of the parameters of the Jaynes-Cummings Hamiltonian. The proposed scheme can be applied to any system that possess the same decomposition of spectrum of the Hamiltonian into a direct sum as in JC model, where eigenenergies in the two direct summands form strictly increasing sequences.
Abstract The “noncommutative graphs” which arise in quantum error correction are a special case of the quantum relations introduced in Weaver (Quantum relations. Mem Am Math Soc 215(v–vi):81–140, 2012). We use this perspective to interpret the Knill–Laflamme error-correction conditions (Knill and Laflamme in Theory of quantum error-correcting codes. Phys Rev A 55:900-911, 1997) in terms of graph-theoretic independence, to give intrinsic characterizations of Stahlke’s noncommutative graph homomorphisms (Stahlke in Quantum zero-error source-channel coding and non-commutative graph theory. IEEE Trans Inf Theory 62:554–577, 2016) and Duan, Severini, and Winter’s noncommutative bipartite graphs (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans Inf Theory 59:1164–1174, 2013), and to realize the noncommutative confusability graph associated to a quantum channel (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans Inf Theory 59:1164–1174, 2013) as the pullback of a diagonal relation. Our framework includes as special cases not only purely classical and purely quantum information theory, but also the “mixed” setting which arises in quantum systems obeying superselection rules. Thus we are able to define noncommutative confusability graphs, give error correction conditions, and so on, for such systems. This could have practical value, as superselection constraints on information encoding can be physically realistic.
Suppressing the effect of decoherence plays a vital role in the theory of quantum information processing.Despite many skepticisms about its feasibility, the existence of error correction schemes in the quantum do- main was proved in the last century, giving the birth to the field of quantum error correction.Since then, the new field has seen remarkably rapid progress in various directions including experimental realizations of quantum error-correcting codes.While quantum information science draws on many branches of physics, computer science, and mathematics, combinatorics is rapidly becoming an indispensable mathematical tool to the study of quantum error correction.In this note, we review some of the latest developments in quantum error correction where combinatorial design theory and coding theory play the central role.
The theory of quantum error correcting codes is a primary tool for fighting decoherence and other quantum noise in quantum communication and quantum computation. Recently, the theory of quantum error correcting codes has developed rapidly and been extended to protect quantum information over asymmetric quantum channels, in which phase-shift and qubit-flip errors occur with different probabilities. In this paper, we generalize the construction of symmetric quantum codes via graphs (or matrices) to the asymmetric case, converting the construction of asymmetric quantum codes to finding matrices with some special properties. We also propose some asymmetric quantum Maximal Distance Separable (MDS) codes as examples constructed in this way.