Time deformations of master equations

TL;DR

This paper investigates how time deformations affect the validity of convolutionless and convolution master equations in open quantum systems, providing criteria to identify non-Markovian behavior through positivity and divisibility properties.

Contribution

It establishes conditions under which time deformations preserve physicality of quantum dynamics, linking these to non-Markovian characteristics of the original system.

Findings

Convolutionless equations remain valid under any time deformation if the dynamics is completely positive divisible.

Uniform time dilations preserve positivity for certain convolution master equations if the original dynamics is positive divisible.

Deformations can serve as witnesses for different degrees of non-Markovian behavior in quantum systems.

Abstract

Convolutionless and convolution master equations are the two mostly used physical descriptions of open quantum systems dynamics. We subject these equations to time deformations: local dilations and contractions of time scale. We prove that the convolutionless equation remains legitimate under any time deformation (results in a completely positive dynamical map) if and only if the original dynamics is completely positive divisible. Similarly, for a specific class of convolution master equations we show that uniform time dilations preserve positivity of the deformed map if the original map is positive divisible. These results allow witnessing different types of non-Markovian behavior: the absence of complete positivity for a deformed convolutionless master equation clearly indicates that the original dynamics is at least weakly non-Markovian; the absence of positivity for a class of time-dilated convolution master equations is a witness of essentially non-Markovian original dynamics.