All maps equivalent to a given map, completely positive or not

TL;DR

This paper characterizes all maps equivalent to a given dynamical map, including non-completely positive ones, revealing that their operator-sum representation freedom is governed by pseudo-unitary transformations.

Contribution

It extends the understanding of operator-sum representations to general maps, identifying the full set of equivalent maps and the nature of their freedom.

Findings

Identifies all maps equivalent to a given dynamical map.

Shows the freedom in OSR for non-completely positive maps is pseudo-unitary.

Provides a unified framework for understanding map equivalence.

Abstract

A dynamical map is a map which takes one density operator to another. Such a map can be written in an operator-sum representation (OSR) using a spectral decomposition. The method of the construction applies to more general maps which need not be completely positive. The OSR not unique; there is a freedom to choose the set of operators in the OSR differently, yet still obtain the same map. Here we identify all maps which are equivalent to a given map. Whereas the freedom for completely positive maps is unitary, the freedom for maps which are not necessarily completely positive is pseudo-unitary.