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Lobachevskii Journal of Mathematics

Operator Random Walks and Quantum Oscillator

Yu. N. Orlov, Vsevolod Zhanovich Sakbaev, Дмитрий Викторович Завадский

Published Apr 1, 2020• 6 citations
Publisher

Abstract

Abstract We investigate asymptotic properties of a sequence whose elements are probability distributions which correspond to compositions of independent random operators. We prove the following statements: 1) a sequence whose elements are the expected values which corresponds to compositions of independent identically distributed random semigroups converges to a semigroup which is generated by the expected value of the random generator; 2) an analog of the law of large numbers for a sequence whose elements are compositions of independent identically distributed random semigroups; 3) a sequence whose elements are compositions of random shift operators on position space converges to a semigroup generated by the Laplace operator; 4) a sequence whose elements are compositions of random shift operators on position space and random shift operators in momentum space converges to a semigroup generated by the Hamiltonian of the quantum oscillator; 5) a sequence whose elements are compositions of random shift operators on phase space converges to a semigroup generated by the Hamiltonian of the quantum oscillator.

DOI
10.1134/s1995080220040186

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We consider quantum random walks in an infinite-dimensional phase space constructed using Weyl representation of the coordinate and momentum operators in the space of functions on a Hilbert space which are square integrable with respect to a shift-invariant measure. We study unitary groups of shift operators in the phase space and averaging of such shifts by Gaussian vectors, which form semigroups of self-adjoint contractions: we find conditions for their strong continuity and establish properties of their generators. Significant differences in their properties allow us to show the absence of the Fourier transform as a unitary transformation that implements the unitary equivalence of these compressive semigroups. Next, we prove the Taylor formula for a certain special subset of smooth functions for shifting to a non-finite vector. It allows us to prove convergence of quantum random walks in the coordinate representation to the evolution of a diffusion process, as well as convergence of quantum random walks in both coordinate and momentum representations to the evolution semigroup of a quantum oscillator in an infinite-dimensional phase space. We find the special essential common domain of generators of semigroups arising in averaging of random shift operators both in position and momentum representations. The invariance of this common domain with respect to both semigroups allows to establish properties of a convex combination of both generators. That convex combination are Hamiltonians of infinite-dimentional quantum oscillators. Thus, we obtain that a Weyl representation of a random walk in an infinite dimensional phase space describes the semigroup of self-adjoint contractions whose generator is the Hamiltonian of an infinite dimensional harmonic oscillator.
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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.
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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We present a procedure for averaging one-parameter random unitary groups and random self-adjoint groups. Central to this is a generalization of the notion of weak convergence of a sequence of measures and the corresponding generalization of the concept of convergence in distribution. The convergence is established in determination of the sequence of compositions of independent random transformations. When sequences of compositions of independent random transformations of the shift by the Euclidean vector in space, the results obtained coincide with the central limit theorem for the sums independent random vectors. The results are applied to the dynamics of quantum systems arising random quantization of the classical Hamiltonian system.