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

Abstract

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.

Classical and quantum part of the environment for quantum Langevin equations

“…In a previous article on one-step evolutions [4], I. Bardet defined the Environment Algebra as the Von Neumann subalgebra of the environment generated by the unitary operator of a one-step evolution on the bipartite system H⊗K. We recall here the basic definitions and the main result on the decomposition of the environment between a maximal commutative and a quantum part.…”

“…For small matrices this can be done for instance numerically. For instance, in [4] it is proved that A(S) ′ is the eigenspace for the eigenvalue 1 of a certain completely positive map on K. This provides by Theorem 2.2 a decomposition of S as S = S 1 + S 2 , such that S 1 is the maximal block-diagonal unitary operator that we can extract from S. However this does not give the decomposition of Φ(K) into a classical and a quantum part. In the next subsection we develop this point with several examples.…”

“…We discuss the notion of change of noise, that is, the effect on the quantum noise of a change of basis. We recall the definition and the main properties of the noise algebra, as defined in [4]. Section 3 is devoted to our main result: a complete characterization of those quantum Langevin equations that give rise to a commutative noise algebra.…”