Diffusion of quantum states generated by a classical random walk

Yu. N. Orlov, V. Zh. Sakbaev

Keldysh Institute of Applied Mathematics of the RAS, Moscow, Russia

Abstract: 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.

Keywords: quantum master equation, quantum state, quantum random walk, Chernoff's theorem, quantum channel.

UDC: 517.983+517.982

DOI: 10.22363/2413-3639-2025-71-2-275-286