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Quantum Dynamical Entropy and non-Markovianity: a collisional model perspective

Giovanni Nichele, Fabio Benatti

Abstract

Accessing the physical mechanisms behind non-Markovian phenomena in open quantum dynamics requires the study of the statistical properties of the joint system-environment dynamics. This is impossible at the level of the reduced dynamics of the open system alone as the latter is obtained by suitably eliminating the environment. The task is instead made possible by considering multi-time correlation functions involving observables of the open system, only: the open system-environment interactions turn them into global ones thus building up correlations between the two systems. Multi-time correlations form the basis of both the theory of quantum stochastic processes and of the Alicki-Lindblad-Fannes dynamical entropy (ALF entropy for short). This latter quantity provides for quantum systems a measure of the dynamical entropy production as the Kolmogorov-Sinai entropy does for classical systems. In the case of a collisional model whereby the dissipative dynamics of a finite-level system is obtained by its coupling to an infinite classical spin chain, the ALF entropy can be explicitly computed. It turns out to depend on the parameters characterizing the statistical properties of the environment and can be related to the activation and super-activation of memory effects in the open quantum system.

Quantum Dynamical Entropy and non-Markovianity: a collisional model perspective

Abstract

Accessing the physical mechanisms behind non-Markovian phenomena in open quantum dynamics requires the study of the statistical properties of the joint system-environment dynamics. This is impossible at the level of the reduced dynamics of the open system alone as the latter is obtained by suitably eliminating the environment. The task is instead made possible by considering multi-time correlation functions involving observables of the open system, only: the open system-environment interactions turn them into global ones thus building up correlations between the two systems. Multi-time correlations form the basis of both the theory of quantum stochastic processes and of the Alicki-Lindblad-Fannes dynamical entropy (ALF entropy for short). This latter quantity provides for quantum systems a measure of the dynamical entropy production as the Kolmogorov-Sinai entropy does for classical systems. In the case of a collisional model whereby the dissipative dynamics of a finite-level system is obtained by its coupling to an infinite classical spin chain, the ALF entropy can be explicitly computed. It turns out to depend on the parameters characterizing the statistical properties of the environment and can be related to the activation and super-activation of memory effects in the open quantum system.
Paper Structure (6 sections, 1 theorem, 41 equations, 2 figures)

This paper contains 6 sections, 1 theorem, 41 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: Quantum Regression and ALF entropy
  3. Collisional models
  4. Quantum Regression
  5. ALF Entropy
  6. Open-system ALF entropy

Key Result

Proposition 1

Let $A \in \mathcal{A}$ and $\omega(A)=\Tr(\rho A)$ for some density matrix $\rho$ and let $\mathcal{X}=\{{X}_a\}_{a=1}^{|\mathcal{X}|}$ be a POVM. Then,

Figures (2)

  • Figure 1: One step $\Theta$ of the algebraic collisional dynamics: (a) $\Phi$ acts on the algebra of the system $\mathcal{A}_S$ and on the $0$-th site of the chain. (b) An operator $A^{[-a,b]}$, localized in $\mathcal{A}_E^{[-a,b]}$, is then translated to the right in (c) by the shift automorphism $\sigma_E$.
  • Figure 2: Dynamical system underlying the construction of the map \ref{['mapdefinition_enunciato']}. (a) $\Theta$ acts non trivially on the $0$-th copy of the system $\mathcal{A}_S^{(0)}=M_d^{(0)}(\mathbb{C})$ and on the algebra of the environment. (b) An operator, initially localized in $\mathcal{A}_S^{[-a,b]}$, is then translated to the right in (c) by the shift automorphism $\sigma_S$. Note the similarity with the algebraic collisional model of Figure \ref{['fig:collisionalfigure']}

Theorems & Definitions (4)

  • Remark 1
  • Remark 2
  • Proposition 1
  • Remark 3