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

TL;DR

This paper generalizes the Gorini-Kossakowski-Sudarshan-Lindblad theorem to include time-dependent generators by exploring the geometry of completely positive maps, using basis-free isomorphisms and finite-dimensional approximations.

Contribution

It extends the GKSL theorem to time-dependent cases and provides a geometric understanding of CP maps without relying on operator algebra representations.

Findings

Generalization of GKSL theorem to time-dependent generators

Finite-dimensional basis-free Choi-Jamiołkowski isomorphism

Kraus decomposition as extremal convex cone decomposition

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-Jamio\l{}kowski 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.