A generalization of the Choi isomorphism with application to open quantum systems

TL;DR

This paper generalizes the Choi isomorphism for quantum transformations using the GKS framework, enabling better analysis of open quantum system dynamics and providing explicit second-order time evolution matrices.

Contribution

It introduces a new GKS-based generalization of the Choi isomorphism, extending its applicability to open quantum systems.

Findings

Derived the GKS matrix for general open quantum system evolution

Extended the Choi isomorphism using GKS formalism

Provided explicit second-order time evolution matrices

Abstract

Completely positive transformations play an important role in the description of state changes in quantum mechanics, including the time evolution of open quantum systems. One useful tool to describe them is the so-called Choi isomorphism, which maps completely positive transformations to positive semi-definite matrices. Accordingly, there are numerous proposals to generalize the Choi isomorphism. In the present paper, we show that the 1976 paper of Gorini, Kossakowski and Sudarshan (GKS) already holds the key for a further generalization and study the resulting GKS isomorphism. As an application, we compute the GKS matrix of the time evolution of a general open quantum system up to second order in time.