Canonical structures of $A$ and $B$ forms

TL;DR

This paper reviews the properties of $A$ and $B$ dynamical maps in finite-dimensional quantum systems, establishing a canonical structure for the $A$ form that succinctly captures the nature of quantum dynamics, with illustrative examples.

Contribution

It introduces a canonical structure for the $A$ form of dynamical maps and proves its equivalence with the $B$ form, clarifying the nature of quantum dynamics.

Findings

Canonical structure of $A$ form captures complete positivity.

Equivalence between $A$ and $B$ forms established.

Illustrated with physical examples of qubit channels.

Abstract

In their seminal paper (Phys. Rev.121, 920 (1961)) Sudarshan, Mathews and Rau investigated properties of the dynamical $A$ and $B$ maps acting on $n$ dimensional quantum systems. Nature of the dynamical maps in open quantum system evolutions has attracted great deal of attention in the later years. However, the novel paper on the $A$ and $B$ dynamical maps has not received its due attention. In this tutorial article we review the properties of $A$ and $B$ forms associated with the dynamics of finite dimensional quantum systems. In particular we investigate a canonical structure associated with the $A$ form and establish its equivalence with the associated $B$ form. We show that the canonical structure of the $A$ form captures the completely positive (not completely positive) nature of the dynamics in a succinct manner. This feature is illustrated through physical examples of qubit channels.