Averaging of random semigroups and quantization

Vsevolod Zhanovich Sakbaev

John Edward Gough, Yurii Nikolaevich Orlov, Vsevolod Zhanovich Sakbaev +1 more

We present a Law of Large Numbers principle for uniformly continuous random quantum dynamical semigroups. Random iterates of independent copies of these semigroups are shown to be Chernoff equivalent to the quantum dynamical semigroup by the average generator.

Ichiro Ohba

Yu. N. Orlov, Vsevolod Zhanovich Sakbaev

We investigate a model that associates random walks in a finite-dimensional Euclidean coordinate space of a classical system with random quantum walks, i.e. random transformations of the set of states of a quantum system arising from quantization of a classical system. As is known, the convolution semigroup of Gaussian measures on a coordinate space admits a representation by a semigroup of self-adjoint contractions in the space of square-integrable functions described by the heat equation. We obtain a representation of the convolution semigroup of Gaussian measures on a coordinate space by a quantum dynamic semigroup in the space of nuclear operators. We give a description of the quantum dynamic semigroup by solutions of the Cauchy problem for a degenerate diffusion equation. We establish the generalized convergence in distribution of a sequence of quantum random walks to an operator-valued random process with values in the Abelian algebra of shift operators by a vector with a normal distribution.

Vladimir Mikhailovich Busovikov, Alexander Nikolaevich Pechen, Vsevolod Zhanovich Sakbaev

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.