Classical and Quantum Parts of the Quantum Dynamics: the Discrete-Time Case

TL;DR

This paper investigates how parts of the environment influence quantum systems in a way that can be considered classical, using algebraic methods to decompose environment actions into classical and quantum components.

Contribution

It introduces the Environment Algebra and Environment Action Algebra to distinguish classical and quantum parts of the environment in open quantum systems, providing a new algebraic framework.

Findings

Decomposition of environment into classical and quantum parts.

Characterization of algebras via spectrum of a CP map.

Finite-dimensional characterization of environment actions.

Abstract

In the study of open quantum systems modeled by a unitary evolution of a bipartite Hilbert space, we address the question of which parts of the environment can be said to have a "classical action" on the system, in the sense of acting as a classical stochastic process. Our method relies on the definition of the Environment Algebra, a relevant von Neumann algebra of the environment. With this algebra we define the classical parts of the environment and prove a decomposition between a maximal classical part and a quantum part. Then we investigate what other information can be obtained via this algebra, which leads us to define a more pertinent algebra: the Environment Action Algebra. This second algebra is linked to the minimal Stinespring representations induced by the unitary evolution on the system. Finally in finite dimension we give a characterization of both algebras in terms of the spectrum of a certain completely positive map acting on the states of the environment.