Convex geometry of Markovian Lindblad dynamics and witnessing non-Markovianity

TL;DR

This paper introduces a geometric framework for detecting non-Markovian quantum dynamics using linear witnesses based on the convex structure of Markovian Choi states, with practical examples for qubit channels.

Contribution

It develops a geometric theory of linear witnesses for non-Markovianity detection, leveraging convex analysis and the structure of Choi states for Lindblad-type evolutions.

Findings

Markovian Choi states form a convex, compact set.

Linear witnesses can separate non-Markovian states from Markovian set.

Markovian Choi states do not form a polytope, enabling nonlinear witness development.

Abstract

We develop a theory of linear witnesses for detecting non-Markovianity, based on the geometric structure of the set of Choi states for all Markovian evolutions having Lindblad type generators. We show that the set of all such Markovian Choi states form a convex and compact set under the small time interval approximation. Invoking geometric Hahn-Banach theorem, we construct linear witnesses to separate a given non-Markovian Choi state from the set of Markovian Choi states. We present examples of such witnesses for dephasing channel and Pauli channel in case of qubits. We further investigate the geometric structure of the Markovian Choi states to find that they do not form a polytope. This presents a platform to consider non-linear improvement of non-Markovianity witnesses.