Canonical form of master equations and characterization of non-Markovianity

TL;DR

This paper introduces a canonical form for master equations in quantum systems, using negative decoherence rates to fully characterize non-Markovianity, and compares this approach with existing measures.

Contribution

It proposes a canonical diagonalization of master equations to uniquely identify non-Markovianity through decoherence rates, enhancing understanding of quantum memory effects.

Findings

Canonical form uniquely characterizes master equations.

Negative decoherence rates fully describe non-Markovianity.

Identifies limitations of existing non-Markovianity measures.

Abstract

Master equations govern the time evolution of a quantum system interacting with an environment, and may be written in a variety of forms. Time-independent or memoryless master equations, in particular, can be cast in the well-known Lindblad form. Any time-local master equation, Markovian or non-Markovian, may in fact also be written in a Lindblad-like form. A diagonalisation procedure results in a unique, and in this sense canonical, representation of the equation, which may be used to fully characterize the non-Markovianity of the time evolution. Recently, several different measures of non-Markovianity have been presented which reflect, to varying degrees, the appearance of negative decoherence rates in the Lindblad-like form of the master equation. We therefore propose using the negative decoherence rates themselves, as they appear in the canonical form of the master equation, to completely characterize non-Markovianity. The advantages of this are especially apparent when more than one decoherence channel is present. We show that a measure proposed by Rivas et al. is a surprisingly simple function of the canonical decoherence rates, and give an example of a master equation that is non-Markovian for all times t>0, but to which nearly all proposed measures are blind. We also give necessary and sufficient conditions for trace distance and volume measures to witness non-Markovianity, in terms of the Bloch damping matrix.