Noether’s theorem for dissipative quantum dynamical semi-groups

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 [J. Math. Phys. 54, 013301 (2013)]. 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 completely positive 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 semi-group.