Noether's Theorem for Dissipative Quantum Dynamical Semigroups

TL;DR

This paper extends Noether's theorem to quantum Markov dynamics, characterizing constants of motion in dissipative quantum systems through fixed points of the Heisenberg semigroup.

Contribution

It provides a quantum analogue of Noether's theorem for dissipative systems, linking constants of motion to fixed points of quantum dynamical semigroups.

Findings

Constants of motion are characterized by fixed points of the Heisenberg semigroup.

The framework applies to finite and infinite-dimensional quantum systems with stationary states.

A mapping from observables to CP maps is constructed to identify conserved quantities.

Abstract

Noether's Theorem on constants of the motion of dynamical systems has recently been extended to classical dissipative systems (Markovian semi-groups) by Baez and Fong. We show how to extend these results to the fully quantum setting of quantum Markov dynamics. For finite-dimensional Hilbert spaces, we construct a mapping from observables to CP maps that leads to the natural analogue of their criterion of commutativity with the infinitesimal generator of the Markov dynamics. Using standard results on the relaxation of states to equilibrium under quantum dynamical semi-groups, we are able to characterise the constants of the motion under quantum Markov evolutions in the infinite-dimensional setting under the usual assumption of existence of a stationary strictly positive density matrix. In particular, the Noether constants are identified with the fixed point of the Heisenberg picture semigroup.