Quantum weak invariants: Dynamical evolution of fluctuations and correlations

TL;DR

This paper investigates the behavior of weak invariants in open quantum systems, revealing their unique role in capturing temporal asymmetry through the evolution of fluctuations, distinct from entropy measures.

Contribution

It introduces a formula for the time evolution of the covariance matrix of weak invariants in systems governed by the GKLS equation, highlighting their dynamical properties.

Findings

Weak invariants' fluctuations grow monotonically over time.

Weak invariants capture temporal asymmetry differently from entropy.

A specific formula for covariance matrix evolution is derived.

Abstract

Weak invariants are time-dependent observables with conserved expectation values. Their fluctuations, however, do not remain constant in time. On the assumption that time evolution of the state of an open quantum system is given in terms of a completely positive map, the fluctuations monotonically grow even if the map is not unital, in contrast to the fact that monotonic increases of both the von Neumann entropy and R\'enyi entropy require the map to be unital. In this way, the weak invariants describe temporal asymmetry in a manner different from the entropies. A formula is presented for time evolution of the covariance matrix associated with the weak invariants in the case when the system density matrix obeys the Gorini-Kossakowski-Lindblad-Sudarshan equation.