Time reversals of irreversible quantum maps

TL;DR

This paper introduces a quantum notion of time reversal for open systems, generalizing classical entropy production and deriving fluctuation relations, with specific constructions for various quantum maps.

Contribution

It defines a quantum time reversal concept for open systems, extending classical ideas, and explores its properties and implications for different types of quantum maps.

Findings

Time reversal in quantum systems can be represented by specific quantum operations.

Entropy production in the environment can be generalized to quantum maps.

Fluctuation relations hold in the quantum setting, similar to classical cases.

Abstract

We introduce the notion of time reversal in open quantum systems as represented by linear quantum operations, and a related generalization of classical entropy production in the environment. This functional is the ratio of the probability to observe a transition between two states under the forward and the time reversed dynamics, and leads, as in the classical case, to fluctuation relations as tautological identities. As in classical dynamics in contact with a heat bath, time reversal is not unique, and we discuss several possibilities. For any bistochastic map its dual map preserves the trace and describes a legitimate dynamics reversed in time, in that case the entropy production in the environment vanishes. For a generic stochastic map we construct a simple quantum operation which can be interpreted as a time reversal. For instance, the decaying channel, which sends the excited state into the ground state with a certain probability, can be reversed into the channel transforming the ground state into the excited state with the same probability.