Quantum Random Walks and Quantum Oscillator in an Infinite-Dimensional Phase Space

TL;DR

This paper develops a mathematical framework for quantum random walks in infinite-dimensional phase space, demonstrating their convergence to quantum oscillator dynamics and analyzing properties of associated semigroups and Hamiltonians.

Contribution

It introduces a novel approach to quantum random walks in infinite-dimensional phase space and characterizes their convergence to quantum oscillator evolution.

Findings

Quantum random walks converge to quantum oscillator dynamics.

Semigroups of shift operators are characterized with their generators.

Hamiltonians of infinite-dimensional oscillators are derived from semigroup properties.

Abstract

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.