Quantum and classical dynamical semigroups of superchannels and semicausal channels

TL;DR

This paper characterizes the generators of dynamical semigroups of superchannels, providing criteria and a normal form analogous to the GKLS form, facilitating analysis and numerical studies of quantum channel decay processes.

Contribution

It introduces a normal form and an efficient criterion for generators of semigroups of superchannels, extending the GKLS framework to superchannels and semicausal maps.

Findings

Normal form for generators of superchannel semigroups derived

Efficient criterion for checking generator validity established

Applicable to infinite-dimensional quantum systems

Abstract

Quantum devices are subject to natural decay. We propose to study these decay processes as the Markovian evolution of quantum channels, which leads us to dynamical semigroups of superchannels. A superchannel is a linear map that maps quantum channels to quantum channels, while satisfying suitable consistency relations. If the input and output quantum channels act on the same space, then we can consider dynamical semigroups of superchannels. No useful constructive characterization of the generators of such semigroups is known. We characterize these generators in two ways: First, we give an efficiently checkable criterion for whether a given map generates a dynamical semigroup of superchannels. Second, we identify a normal form for the generators of semigroups of quantum superchannels, analogous to the GKLS form in the case of quantum channels. To derive the normal form, we exploit the relation between superchannels and semicausal completely positive maps, reducing the problem to finding a normal form for the generators of semigroups of semicausal completely positive maps. We derive a normal for these generators using a novel technique, which applies also to infinite-dimensional systems. Our work paves the way to a thorough investigation of semigroups of superchannels: Numerical studies become feasible because admissible generators can now be explicitly generated and checked. And analytic properties of the corresponding evolution equations are now accessible via our normal form.