  Quantum error correction plays a key role for quantum information
transmission and quantum computing. In this work, we develop and apply the
theory of non-commutative 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 Hamiltonian with a bosonic coherent field. We extend the theory
of non-commutative graphs to this situation and construct, using the
Gazeau-Klauder coherent states, the corresponding non-commutative 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.
