Symmetries Versus Conservation Laws in Dynamical Quantum Systems: A Unifying Approach Through Propagation of Fixed Points

Abstract

We unify recent Noether-type theorems on the equivalence of symmetries with conservation laws for dynamical systems of Markov processes, of quantum operations, and of quantum stochastic maps, by means of some abstract results on propagation of fixed points for completely positive maps on C *-algebras. We extend most of the existing results with characterisations in terms of dual infinitesimal generators of the corresponding strongly continuous one-parameter semigroups. By means of an ergodic theorem for dynamical systems of completely positive maps on von Neumann algebras, we show the consistency of the condition on the standard deviation for dynamical systems of quantum operations, and hence of quantum stochastic maps as well, in case the underlying Hilbert space is infinite dimensional.

Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states

“…The first statement is regarding the relationship between the conserved quantities J and the Kraus of operators of E. It is a generalization of a theorem for fixed points of faithful channels [24,[61][62][63][64], which states that a conserved quantity J with eigenvalue ∆ = 0 commutes with all of the Kraus operators. It is shown that conserved quantities with ∆ = 0 commute up to a phase.…”

Absorption and Fixed Points for Semigroups of Quantum Channels

“…Another topic which has drown attention in the last years is the study of the fixed points of the evolution: they are relevant for the asymptotics of the evolution of quantum systems ( [10]) and whether or not they are a W * -algebra has implications, for instance, on the relationship between conserved quantities and symmetries of the semigroup (see [2] about the general problem of when fixed points are an algebra and [12] for a discussion about Noether-type results in the context of quantum channels). Fixed points are well understood in the case of positive recurrent semigroups ([4, 9, 14]), while less is known for general semigroups ( [1,3,11]).…”