Quantum Random Walks and Time Reversal

TL;DR

This paper introduces a framework for quantum random walks based on Hopf algebras, providing a physical interpretation, a representation theorem, and a duality concept that relates to time reversal in quantum systems.

Contribution

It develops a representation theorem for quantum random walks associated with Hopf algebras and introduces a duality operation linked to time reversal in quantum processes.

Findings

Representation of Hopf algebra in linear operators with a time evolution operator W

Duality operation for quantum random walks and diffusions

Introduction of a coentropy concept and a CTP-type theorem

Abstract

Classical random walks and Markov processes are easily described by Hopf algebras. It is also known that groups and Hopf algebras (quantum groups) lead to classical and quantum diffusions. We study here the more primitive notion of a quantum random walk associated to a general Hopf algebra and show that it has a simple physical interpretation in quantum mechanics. This is by means of a representation theorem motivated from the theory of Kac algebras: If $H$ is any Hopf algebra, it may be realised in $\Lin(H)$ in such a way that $\Delta h=W(h\tens 1)W^{-1}$ for an operator $W$. This $W$ is interpreted as the time evolution operator for the system at time $t$ coupled quantum-mechanically to the system at time $t+\delta$. Finally, for every Hopf algebra there is a dual one, leading us to a duality operation for quantum random walks and quantum diffusions and a notion of the coentropy of an observable. The dual system has its time reversed with respect to the original system, leading us to a CTP-type theorem.