Large deviations for quantum Markov semigroups on the 2 x 2 matrix algebra

TL;DR

This paper establishes a large deviation principle for quantum Markov semigroups on 2x2 matrices with an absorbing pure state, showing convergence properties and independence of initial states, with applications in weak coupling limits.

Contribution

It proves a large deviation principle for quantum Markov semigroups on 2x2 matrices with a pure state, linking the rate function to the generator and demonstrating asymptotic faithfulness.

Findings

Large deviation principle established for the state evolution.

Rate function expressed in terms of the generator.

States become faithful for large times regardless of initial state.

Abstract

Let $({\mathcal{T}}_{*t})$ be a predual quantum Markov semigroup acting on the full 2 x 2 matrix algebra and having an absorbing pure state. We prove that for any initial state $\omega$, the net of orthogonal measures representing the net of states $({\mathcal{T}}_{*t}(\omega))$ satisfies a large deviation principle in the pure state space, with a rate function given in terms of the generator, and which does not depend on $\omega$. This implies that $({\mathcal{T}}_{*t}(\omega))$ is faithful for all $t$ large enough. Examples arising in weak coupling limit are studied.