Non-commutative graphs and quantum error correction for a two-mode quantum oscillator

TL;DR

This paper extends the theory of non-commutative graphs to infinite-dimensional quantum systems, specifically analyzing error correction for a two-mode quantum oscillator, and identifies maximal quantum anticliques.

Contribution

It introduces a novel application of non-commutative graphs to infinite-dimensional Hilbert spaces in quantum error correction, with explicit examples for a two-mode oscillator.

Findings

Constructed non-commutative graphs for a two-mode quantum oscillator

Identified maximal quantum anticliques in these graphs

Demonstrated applicability to practical quantum systems

Abstract

An important topic in quantum information is the theory of error correction codes. Practical situations often involve quantum systems with states in an infinite dimensional Hilbert space, for example coherent states. Motivated by these practical needs, we apply the theory of non-commutative graphs, which is a tool to analyze error correction codes, to infinite dimensional Hilbert spaces. As an explicit example, a family of non-commutative graphs associated with the Schr\"odinger equation describing the dynamics of a two-mode quantum oscillator is constructed and maximal quantum anticliques for these graphs are found.