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The Gorini-Kossakowski-Sudarshan-Lindblad problem and the geometry of CP maps

Paul E. Lammert

TL;DR

This work develops a geometric, basis-free framework for the GKSL problem by leveraging a generalized Jamiołkowski isomorphism that links completely positive maps to positive operators. It establishes Kraus decompositions as extremal structures within CP maps, and introduces an L-cone and Lindblad parametrizers to characterize time-dependent, Markovian CP evolution in both finite and separable settings. The approach avoids operator-algebraic representation theory, instead using filtrations, the d-metric, and strong/weak operator topologies to bootstrap finite-dimensional results to the separable case. The paper also clarifies the categorical and functorial relationships (via the $\bm{\theta}$ map) between closed and open quantum systems, and provides dimension-independent bounds and constructive procedures for obtaining Kraus decompositions and Lindblad parametrizations. Overall, it delivers a coherent, generalizable methodology for understanding CP dynamics in infinite-dimensional quantum systems with practical implications for modeling open-system evolution.

Abstract

The Lindblad equation embodies a fundamental paradigm of the quantum theory of open systems, and the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generation theorem says precisely which superoperators can appear on its right-hand side. These are the generators of completely positive trace-preserving (or nonincreasing) semigroups. We prove a generalization, with time-dependent generator, as an application of an investigation of the geometry of the class of completely positive (CP) maps. The treatment of the finite-dimensional setting is based on a basis-free Choi-Jamiołkowski type isomorphism. The infinite-dimensional case is bootstrapped from the finite-dimensional theory via a sequence of finite-dimensional approximations. Kraus decomposition is established along the way, in the guise of an extremal decomposition of the closed convex cone of CP maps. No appeal is made to results from the representation theory of operator algebras.

The Gorini-Kossakowski-Sudarshan-Lindblad problem and the geometry of CP maps

TL;DR

This work develops a geometric, basis-free framework for the GKSL problem by leveraging a generalized Jamiołkowski isomorphism that links completely positive maps to positive operators. It establishes Kraus decompositions as extremal structures within CP maps, and introduces an L-cone and Lindblad parametrizers to characterize time-dependent, Markovian CP evolution in both finite and separable settings. The approach avoids operator-algebraic representation theory, instead using filtrations, the d-metric, and strong/weak operator topologies to bootstrap finite-dimensional results to the separable case. The paper also clarifies the categorical and functorial relationships (via the map) between closed and open quantum systems, and provides dimension-independent bounds and constructive procedures for obtaining Kraus decompositions and Lindblad parametrizations. Overall, it delivers a coherent, generalizable methodology for understanding CP dynamics in infinite-dimensional quantum systems with practical implications for modeling open-system evolution.

Abstract

The Lindblad equation embodies a fundamental paradigm of the quantum theory of open systems, and the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generation theorem says precisely which superoperators can appear on its right-hand side. These are the generators of completely positive trace-preserving (or nonincreasing) semigroups. We prove a generalization, with time-dependent generator, as an application of an investigation of the geometry of the class of completely positive (CP) maps. The treatment of the finite-dimensional setting is based on a basis-free Choi-Jamiołkowski type isomorphism. The infinite-dimensional case is bootstrapped from the finite-dimensional theory via a sequence of finite-dimensional approximations. Kraus decomposition is established along the way, in the guise of an extremal decomposition of the closed convex cone of CP maps. No appeal is made to results from the representation theory of operator algebras.

Paper Structure

This paper contains 82 sections, 32 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Generalized Jamiołkowski isomorphism: the power of swap
  3. Notational conventions
  4. Spaces of operators
  5. Dual space
  6. Hilbert-Schmidt space
  7. Jamiołkowski transformation
  8. A fundamental isomorphism
  9. Jamiołkowski transformation
  10. Key properties of $\mathfrak{J}$
  11. Operational significance of the Hilbert-Schmidt inner product
  12. Jamiołkowski correspondence between classes of operators
  13. More spaces of operators
  14. Hermitian operators $\mathrm{SA}(\mathcal{H})$
  15. Positive operators $\mathrm{Pos}(\mathcal{H})$
  16. ...and 67 more sections

Key Result

Proposition 3.1

The class of $\mathsf{CP}$ maps is the unique class satisfying the $\mathit{OK}$ postulates.

Theorems & Definitions (99)

  • Definition 2.1
  • proof : Proof of (\ref{['eq:own inverse']})
  • proof : Proof of (\ref{['eq:jam rank 1']})
  • proof : Proof of (\ref{['eq:jam distributes over tensor']})
  • Example 2.1: Orthonormal bases
  • Definition 2.2: ${\bm{\pi}}$
  • Definition 3.1: ${\bm{\theta}}$
  • Definition 3.2: convex set
  • Definition 3.3: cones
  • Example 3.1: $\mathrm{Pos}(\mathcal{K})$ is a proper cone in $\mathrm{SA}(\mathcal{K})$
  • ...and 89 more