Maps and inverse maps in open quantum dynamics

TL;DR

This paper investigates different types of maps describing open quantum system dynamics, establishing conditions for their invertibility, and highlighting differences between maps based on fixed mean values and correlations.

Contribution

It provides a comprehensive analysis of inverse conditions for affine maps in open quantum dynamics, including criteria for complete positivity and the independence of inverse maps from the larger system's dynamics.

Findings

Inverse maps exist iff the homogeneous part is invertible.

Some maps are not completely positive, but their homogeneous parts are.

Inverse maps are generally not determined solely by the larger system's unitary dynamics.

Abstract

Two kinds of maps that describe evolution of states of a subsystem coming from dynamics described by a unitary operator for a larger system, maps defined for fixed mean values and maps defined for fixed correlations, are found to be quite different for the same unitary dynamics in the same situation in the larger system. An affine form is used for both kinds of maps to find necessary and sufficient conditions for inverse maps. All the different maps with the same homogeneous part in their affine forms have inverses if and only if the homogeneous part does. Some of these maps are completely positive; others are not, but the homogeneous part is always completely positive. The conditions for an inverse are the same for maps that are not completely positive as for maps that are. For maps defined for fixed mean values, the homogeneous part depends only on the unitary operator for the dynamics of the larger system, not on any state or mean values or correlations. Necessary and sufficient conditions for an inverse are stated several different ways: in terms of the maps of matrices, basis matrices, density matrices, or mean values. The inverse maps are generally not tied to the dynamics the way the maps forward are. A trace-preserving completely positive map that is unital can not have an inverse that is obtained from any dynamics described by any unitary operator for any states of a larger system.