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Lobachevskii Journal of Mathematics

Chernoff Equivalence for Shift Operators, Generating Coherent States in Quantum Optics

Leonid Andreevich Borisov, Yu. N. Orlov, V. Zh. Sakbaev

Published Aug 1, 2018• 8 citations
Publisher

Abstract

In this paper some examples of sequences of compositions of n independent equally distributed random semigroups in the form of a shift operator generating coherent states in quantum optics have been studied. Representation of averaged coherent states in quantum optics using a procedure of averaging of random shift operators is obtained. The analytical expression for averaged value of a superposition of shift operators is presented.

DOI
10.1134/s1995080218060033

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In this paper the random channels and their compositions in the space of quantum states are studied. For compositions of i.i.d. random unitary channels, the limit behaviour of probability distributions is described. The sufficient condition for convergence in probability is obtained. The generalized convergence in distribution w.r.t. weak operator topology is obtained. The analysis of transmission of pure and normal states to the set of singular states is done. The dynamics of quantum states is described in terms of the evolution of the values of quadratic forms of operators from the algebra that implements the representation of canonical commutation relations.
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Abstract Random walks in a real separable Hilbert space correspond to a composition of random unitary operators in the space of functions square integrable with respect to a translation-invariant measure, as well as to a composition of random unitary channels in the space of quantum states. Conditions on the distribution of random walks that are necessary and sufficient for the continuity of the mathematical expectations of their representations in the Hilbert space and the space of quantum states are obtained. The dynamics of the set of quantum states obtained by averaging random shifts over the semigroup (with respect to the convolution operation) of centered Gaussian measures on the Hilbert space is studied. It is shown that, depending on the property of the covariance operator of the Gaussian measure to have a nuclear square root, such dynamics is either a strongly continuous quantum dynamical semigroup or instantaneously destroys the set of normal states. In the latter case, the dynamics of singular quantum states is described in terms of the evolution of mean values on an operator algebra that implements a representation of canonical commutation relations.