Quantum Dynamical Entropy and Dissipative Information Flows

Abstract

The Alicki-Lindblad-Fannes dynamical (ALF) entropy measures the rate at which new information is gathered about a quantum system by inspecting its long-time evolution. We propose an extension of the ALF entropy to open quantum dynamics as a measure of back-flow of information from the environment. Such a proposal is stronger than the existing ones based only on the open system reduced dynamics. In the case of a qubit collisionally coupled to a classical spin chain, we obtain an exact expression for the $\textit{open-system ALF entropy}$ explicitly depending on the environment correlations. An extreme case shows how the information flow from environment to system corresponds to vanishing entropy production as for reversible finite quantum systems.

Figures (2)

• Figure 1: (a) One step $\Theta$ of the Heisenberg collisional dynamics: $\Phi$ jointly acts on the system observables in $M_d\equiv M_d(\mathbb{C})$ and on those of the $0$-th site of the chain. (b) One step of the dynamics for the open quantum chain $M_d^{\mathbb{Z}}(\mathbb{C})$: at each tick of time, the system copy at the 0-th site interacts with the environment $E$, here represented as an infinite spin chain.
• Figure 2: Open-system dynamical entropy \ref{['specificexample']} as a function of $\Delta/p$ for fixed values $r=0.1$ and $p=0.25$. The colored regions correspond to different divisibility degrees of the reduced dynamics $\Lambda_n$. For $\Delta/p\to 1$, $\Lambda_n$ is not P-divisible and shows distinguishability revivals; in the pink region, $\Lambda_n$ becomes P-divisible and does not show revivals, though they can superactivate for $\Lambda_n\otimes\Lambda_n$. The latter becomes P-divisible in the red region, while in the dark-red region $\Delta/p\ll 1$, CP-divisibility is achieved.