Thermodynamics of linear open quantum walks

TL;DR

The paper develops a complete thermodynamic description for linear open quantum walks, defining an equilibrium temperature and showing a finite-$ω$ population inversion at $ω_c=1/2$. By mapping steady states to Boltzmann distributions with a linear energy spectrum, it derives the partition function, internal energy, entropy, and heat capacity, revealing two distinct large-$N$ regimes and a symmetry $S(ω)=S(1-ω)$. It further analyzes nonequilibrium thermodynamics, providing bounds and analytic approximations for thermalization times and entropy evolution, and demonstrates positive entropy production in line with the second law. The work connects thermodynamic constraints to dissipative quantum computation, offering practical tools for reservoir engineering and insights into energy costs for control of the environment-driven quantum walks.

Abstract

Open quantum systems interact with their environment, leading to nonunitary dynamics. We investigate the thermodynamics of linear Open Quantum Walks (OQWs), a class of quantum walks whose dynamics is entirely driven by the environment. We define an equilibrium temperature, identify a population inversion near a finite critical value of a control parameter, analyze the thermalization process, and develop the statistical mechanics needed to describe the thermodynamical properties of linear OQWs. We also study nonequilibrium thermodynamics by analyzing the time evolution of entropy, energy, and temperature, while providing analytical tools to understand the system's evolution as it converges to the thermalized state. We examine the validity of the second and third laws of thermodynamics in this setting. Finally, we employ these developments to shed light on dissipative quantum computation within the OQW framework.

Figures (12)

• Figure 1: An arbitrary open quantum walk can be represented by this visual diagram. If there is an omitted edge in a particular diagram, this means that the corresponding operator $B_i^j$ is zero. Figure extracted from Ref. linck25.
• Figure 2: The diagram corresponding to the linear OQW model. Each $U_i$ is a unitary operator, and $\omega, \lambda \geq 0$ are such that $\omega + \lambda = 1$. Figure extracted from Ref. linck25.
• Figure 3: Plot of the steady-state probability as a function of the position in the graph for $\omega = 1/3, \; 1/2, \; 2/3$ and fixed $N = 30$.
• Figure 4: Temperature of a linear OQW as a function of $\omega$ for $\varepsilon = 1$. The population inversion can be seen as the parameter $\omega$ crosses the divergence in $\omega = 1/2$.
• Figure 5: Entropy versus $\beta$ in the linear OQW for $N = 500$.