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arXiv:quant-ph/0406213 · Journal of Physics A Mathematical and General

A pathwise ergodic theorem for quantum trajectories

Burkhard Kümmerer, Hans Maassen

Published Nov 25, 2004• 56 citations
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Abstract

If the time evolution of an open quantum system approaches equilibrium in the time mean, then on any single trajectory of any of its unravellings the time-averaged state approaches the same equilibrium state with probability 1. In the case of multiple equilibrium states, the quantum trajectory converges in the mean to a random choice from these states.

DOI
10.1088/0305-4470/37/49/008

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If the time evolution of an open quantum system approaches equilibrium in the time mean, then on any single trajectory of any of its unravellings the time-averaged state approaches the same equilibrium state with probability 1. In the case of multiple equilibrium states, the quantum trajectory converges in the mean to a random choice from these states.
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von Neumann has proved a quantum-mechanical ergodic theorem which deals with a closed macroscopic system and macroscopic observables. Certain unsatisfactory features of this theorem are probably due to the overly-strong idealization of a completely closed system.Since the claims of statistical mechanics go beyond macroscopic systems, there is a need for a more general theorem. The present paper deals with a system of any number of degrees of freedom and with any observable, during interaction with a temperature bath. In order to avoid an assumption of randomness, the bath must be described explicitly and precisely by a time-independent Hamiltonian. The model of the bath is obtained by taking the Gibbs ensemble seriously, i.e., as a set of NNN identical systems interacting through a potential $\ensuremath{\lambda}V$.The following theorem is proved. The time average of the quantum-mechanical expectation value of any observable with respect to any initial state is equal to its statistical average, in the double limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\lambda}\ensuremath{\rightarrow}0$, for the overwhelming majority of all interaction potentials VVV.
Abstract We consider quantum trajectories arising from disordered, repeated generalized measurements, which have the structure of Markov chains in random environments (MCREs) with dynamically-defined transition probabilities; we call these disordered quantum trajectories . Under the assumption that the underlying disordered open quantum dynamical system approaches a unique equilibrium in time averages, we establish a strong law of large numbers for measurement outcomes arising from disordered quantum trajectories, which follows after we establish general annealed ergodic theorems for the corresponding MCRE. The type of disorder our model allows includes the random settings where the disorder is i.i.d. or Markovian, the periodic (resp. quasiperiodic) settings where the disorder has periodic (resp. quasiperiodic) structure, and the nonrandom setting where the disorder is constant through time. In particular, our work extends the earlier noise-free results of Kümmerer and Maassen to the present disordered framework.
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