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.

• Received 24 December 2020
• Accepted 17 March 2021

DOI:https://doi.org/10.1103/PhysRevA.103.042407