Relative purity, speed of fluctuations, and bounds on equilibration times

TL;DR

This paper investigates the local equilibration process in closed quantum systems using relative purity, deriving bounds on fluctuation sizes, fluctuation speeds, and equilibration times, with implications for quantum speed limits and metrology.

Contribution

It introduces new bounds on fluctuations and equilibration times based on relative purity, linking quantum coherences and correlations to the speed of equilibration.

Findings

Bound on average fluctuation size depending on effective dimension

Relation between fluctuation speed and quantum coherences

Lower bounds on equilibration time derived

Abstract

We discuss the local equilibration of closed systems using the relative purity, a paradigmatic information-theoretic distinguishability measure that finds applications ranging from quantum metrology to quantum speed limits. First we obtain an upper bound on the average size of the fluctuations on the relative purity: it depends on the effective dimension resembling the bound obtained with the trace distance. Second, we investigate the dynamics of relative purity and its rate of change as a probe of the speed of fluctuations around the equilibrium. In turn, such speed captures the notion of how fast some nonequilibrium state approaches the steady state under the local nonunitary dynamics, somehow giving the information of the quantum speed limit towards the equilibration. We show that the size of fluctuations depends on the quantum coherences of the initial state with respect to the eigenbasis of the Hamiltonian, also addressing the role played by the correlations between system and reservoir into the averaged speed. Finally, we have derived a family of lower bounds on the time of evolution between these states, thus obtaining an estimate for the equilibration time at the local level. These results could be of interest to the subjects of equilibration, quantum speed limits, and also quantum metrology.