CP conditions for GKSL-like master equations

TL;DR

This paper derives conditions under which quantum dynamical maps with GKSL-like master equations, including time-dependent and non-local cases, are guaranteed to be completely positive, extending the understanding of non-Markovian quantum dynamics.

Contribution

It provides rigorous CP conditions for GKSL-like master equations with arbitrary time dependence, including time-local, time-non-local, and approximated forms.

Findings

Derived CP conditions for time-dependent GKSL-like MEs.

Extended CP analysis to time-non-local and approximated cases.

Clarified the relation between CP and non-Markovian dynamics.

Abstract

The complete positivity (CP) of a quantum dynamical map (QDM) is, in general, difficult to show when its master equation (ME) does not conform to the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form. The GKSL ME describes the Markovian dynamics, comprising a unitary component with time-independent Hermitian operators and a non-unitary component with time-independent Lindblad operators and positive time-independent damping rates. Recently, the non-Markovian dynamics has received growing attention, and the various types of GKSL-like MEs with time-dependent operators are widely discussed; however, rigorous discussions on their CP conditions remain limited. This paper presents conditions for QDMs to be CP, whose MEs take the GKSL-like form with arbitrary time dependence. One case considered is where its ME takes the time-local integro-differential GKSL-like form, which includes CP-divisible cases. Another case considered is where the ME is time-non-local but can be approximated to be time-local in the weak-coupling regime. As a special case of the time-non-local case, the same discussion holds for the time-convoluted GKSL-like form, which should be compared to previous studies.