Thermodynamics of N-dimensional quantum walks

TL;DR

This paper develops a thermodynamic framework for N-dimensional quantum walks by linking entanglement entropy to thermodynamic quantities, and analyzes the 2D case with different initial states.

Contribution

It introduces a novel thermodynamic approach to quantum walks based on entanglement, providing explicit formulas for thermodynamic variables from the reduced density matrix.

Findings

Entanglement entropy can be interpreted thermodynamically.

The entanglement temperature depends on initial conditions and system parameters.

Explicit analysis for 2D quantum walks with different initial states.

Abstract

The entanglement between the position and coin state of a $N$-dimensional quantum walker is shown to lead to a thermodynamic theory. The entropy, in this thermodynamics, is associated to the reduced density operator for the evolution of chirality, taking a partial trace over positions. From the asymptotic reduced density matrix it is possible to define thermodynamic quantities, such as the asymptotic entanglement entropy, temperature, Helmholz free energy, etc. We study in detail the case of a $2$-dimensional quantum walk, in the case of two different initial conditions: a non-separable coin-position initial state, and a separable one. The resulting entanglement temperature is presented as function of the parameters of the system and those of the initial conditions.