We study aspects of the thermodynamics of quantum versions of spin glasses. By means of the Lie-Trotter formula for exponential sums of operators, we adapt methods used to analyze classical spin glass models to answer analogous questions about quantum models.Comment: 17 page

We consider a lattice of weakly interacting quantum Markov processes. Without interaction, the dynamics at each site is relaxing exponentially to a unique stationary state. With interaction, we show that there remains a unique stationary state in the thermodynamic limit, i.e. absence of phase coexistence, and the relaxation towards it is exponentially fast for local observables. We do not assume that the quantum Markov process is reversible (detailed balance) w.r.t. a local Hamiltonian.

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The quantum theory of indirect measurements in physical systems is studied. The example of an indirect measurement of an observable represented by a self-adjoint operator N with finite spectrum is analysed in detail. The Hamiltonian generating the time evolution of the system in the absence of direct measurements is assumed to be given by the sum of a term commuting with N and a small perturbation not commuting with N . The system is subject to repeated direct (projective) measurements using a single instrument whose action on the state of the system commutes with N . If the Hamiltonian commutes with the observable N (i.e., if the perturbation vanishes) the state of the system approaches an eigenstate of N , as the number of direct measurements tends to ∞. If the perturbation term in the Hamiltonian does not commute with N the system exhibits "jumps" between different eigenstates of N . We determine the rate of these jumps to leading order in the strength of the perturbation and show that if time is re-scaled appropriately a maximum likelihood estimate of N approaches a Markovian jump process on the spectrum of N , as the strength of the perturbation tends to 0.

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