On the construction of a quantum channel corresponding to non-commutative graph for a qubit interacting with quantum oscillator
G. G. Amosov, A. S. Mokeev, A. N. Pechen
TL;DR
The paper addresses error correction for a qubit interacting with a quantum oscillator by employing the theory of non-commutative operator graphs generated by POVMs. It develops a generalized quantum channel between preduals of von Neumann algebras, using a density P(ω) to define a POVM-derived graph V = overline{span}{M(B)} and its complementary channel, thereby linking POVMs, Naimark dilation, and graph-based quantum error correction. For the qubit–oscillator model, the authors provide an explicit partition of the Hilbert space and construct Gauzeau– Klauder coherent-state-based POVMs M that are generated by unitary dynamics, yielding a quantum anticlique P3 that identifies the error-correcting subspace. This framework clarifies how infinite-dimensional systems can be analyzed through generalized channels and graphs, informing robust error correction in quantum information processing.
Abstract
We consider error correction, based on the theory of non-commutative graphs, for a model of a qubit interacting with quantum oscillator. The dynamics of the composite system is governed by the Schrödinger equation which generates positive operator-valued measure (POVM) for the system dynamics. We construct a quantum channel generating the non-commutative graph as a linear envelope of the POVM. The idea is based on applying a generalized version of a quantum channel using the apparatus of von Neumann algebras. The results are analyzes for a non-commutative graph generated by a qubit interacting with quantum oscillator. For this model the quantum anticlique which determines the error correcting subspace has an explicit expression.