A Pathwise Ergodic Theorem for Quantum Trajectories

If the time evolution of an open quantum system approaches equilibrium in the time mean, then on any single trajectory of any of its unravelings 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.Comment: 8 page

The Free Quon Gas Suffers Gibbs' Paradox

We consider the Statistical Mechanics of systems of particles satisfying the $q$-commutation relations recently proposed by Greenberg and others. We show that although the commutation relations approach Bose (resp.\ Fermi) relations for $q\to1$ (resp.\ $q\to-1$), the partition functions of free gases are independent of $q$ in the range $-1<q<1$. The partition functions exhibit Gibbs' Paradox in the same way as a classical gas without a correction factor $1/N!$ for the statistical weight of the $N$-particle phase space, i.e.\ the Statistical Mechanics does not describe a material for which entropy, free energy, and particle number are extensive thermodynamical quantities.Comment: number-of-pages, LaTeX with REVTE

Rigorous mean field model for CPA: Anderson model with free random variables

A model of a randomly disordered system with site-diagonal random energy fluctuations is introduced. It is an extension of Wegner's $n$-orbital model to arbitrary eigenvalue distribution in the electronic level space. The new feature is that the random energy values are not assumed to be independent at different sites but free. Freeness of random variables is an analogue of the concept of independence for non-commuting random operators. A possible realization is the ensemble of at different lattice-sites randomly rotated matrices. The one- and two-particle Green functions of the proposed hamiltonian are calculated exactly. The eigenstates are extended and the conductivity is nonvanishing everywhere inside the band. The long-range behaviour and the zero-frequency limit of the two-particle Green function are universal with respect to the eigenvalue distribution in the electronic level space. The solutions solve the CPA-equation for the one- and two-particle Green function of the corresponding Anderson model. Thus our (multi-site) model is a rigorous mean field model for the (single-site) CPA. We show how the Llyod model is included in our model and treat various kinds of noises.Comment: 24 pages, 2 diagrams, Rev-Tex. Diagrams are available from the authors upon reques

Elements of Quantum Probability

this article during a stay at the \Mathematisches Forschungsinstitut Oberwolfach&quot;, nanced also from the European Human Capital and Mobility Program on Quantum Probability. nite coin tosses in the third section. Suprisingly, they lead quite quickly to recent developments. In an appendix we indicate the steps which lead to the full mathematical model of quantum probability. 1. Why Classical Probability does not Suffic

Invariant Measure for Quantum Trajectories

International audienceWe study a class of Markov chains that model the evolution of a quantum system subject to repeated measurements. Each Markov chain in this class is defined by a measure on the space of matrices, and is then given by a random product of correlated matrices taken from the support of the defining measure. We give natural conditions on this support that imply that the Markov chain admits a unique invariant probability measure. We moreover prove the geometric convergence towards this invariant measure in the Wasserstein metric. Standard techniques from the theory of products of random matrices cannot be applied under our assumptions, and new techniques are developed, such as maximum likelihood-type estimations