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

Heinz-Jürgen Schmidt

TL;DR

This work generalizes the Choi isomorphism through the Gorini–Kossakowski–Sudarshan (GKS) formalism by introducing a GKS matrix $g$ associated with an arbitrary orthonormal basis in ${\mathcal A}=L({\mathcal H})$, establishing that complete positivity of a map ${\mathcal E}$ is equivalent to $g\ge 0$. The Choi isomorphism emerges as a special case of the GKS framework, and the paper analyzes how GKS matrices transform under basis changes and how trace preservation and Hermiticity are encoded in $g$. It then applies the GKS formalism to open-system dynamics, deriving the time-dependent GKS equation and its reduction to the time-dependent Lindblad equation, and performs a perturbative expansion up to $O(t^2)$ to show positivity of the GKS matrix in that regime. The study also compares GKS with other generalized Choi-type isomorphisms (DJ, PSKH, FC), clarifying their relations and distinctions, and demonstrates the practical relevance of the GKS approach for open quantum systems and CP maps.

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.

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

TL;DR

This work generalizes the Choi isomorphism through the Gorini–Kossakowski–Sudarshan (GKS) formalism by introducing a GKS matrix associated with an arbitrary orthonormal basis in , establishing that complete positivity of a map is equivalent to . The Choi isomorphism emerges as a special case of the GKS framework, and the paper analyzes how GKS matrices transform under basis changes and how trace preservation and Hermiticity are encoded in . It then applies the GKS formalism to open-system dynamics, deriving the time-dependent GKS equation and its reduction to the time-dependent Lindblad equation, and performs a perturbative expansion up to to show positivity of the GKS matrix in that regime. The study also compares GKS with other generalized Choi-type isomorphisms (DJ, PSKH, FC), clarifying their relations and distinctions, and demonstrates the practical relevance of the GKS approach for open quantum systems and CP maps.

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.
Paper Structure (22 sections, 9 theorems, 103 equations, 1 figure)

This paper contains 22 sections, 9 theorems, 103 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and general results
  3. The Choi isomorphism
  4. The GKS isomorphism
  5. Choi isomorphism as a special case of the GKS isomorphism
  6. Definition of the GKS isomorphisms
  7. Properties of the GKS isomorphisms
  8. Comparison with other isomorphisms
  9. Comparison with dePillis-Jamiołski isomorphism
  10. Comparison with Paulsen-Shultz-Kye-Han isomorphism
  11. Comparison with Frembs-Cavalcanti isomorphism
  12. Application to the time evolution of open quantum systems
  13. Time-dependent GKS isomorphism
  14. Time-dependent Lindblad equation
  15. Expansion up to $O(t^2)$ terms
  16. ...and 7 more sections

Key Result

Proposition 1

Figures (1)

  • Figure 1: Illustration of the Choi matrix isomorphism $c({\mathcal{E}})$ for the case of $N=2$. It performs a permutation of the matrix elements of the $4\times 4$ matrix ${\mathcal{E}}$, which consists of four transpositions marked by red double arrows. The matrix elements in the gray squares remain fixed.

Theorems & Definitions (9)

  • Proposition 1
  • Theorem 1
  • Lemma 1
  • Proposition 2
  • Lemma 2
  • Proposition 3
  • Theorem 2
  • Theorem 3
  • Theorem 4