Thermodynamics of linear open quantum walks

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.

💡 Research Summary

The paper presents a comprehensive thermodynamic framework for linear Open Quantum Walks (OQWs), a class of quantum walks whose dynamics is entirely driven by interaction with an external environment. After a brief introduction to quantum thermodynamics and quantum walks, the authors review the mathematical formalism of OQWs, emphasizing the Kraus‑operator representation that guarantees complete positivity and trace preservation. They specialize to a one‑dimensional lattice with nearest‑neighbour jumps: a rightward jump occurs with probability ω (accompanied by a unitary U_i) and a leftward jump with probability λ=1−ω (accompanied by a complementary unitary). The transition matrix T thus defines a non‑symmetric Markov chain whose stationary distribution is π_m∝a^m with a=ω/λ.

Interpreting this stationary distribution as a Boltzmann distribution, the authors assign linear energy levels E_m=mε (ε>0) to the lattice sites. This yields a temperature definition T=−ε/(k_B ln (ω/λ)). At the critical point ω_c=½ the temperature diverges; for ω>½ the inverse temperature β becomes negative, indicating a population inversion (higher‑energy sites become more populated). The paper derives explicit expressions for the partition function Z, mean energy ⟨E⟩, energy fluctuations, entropy S(β)=ln Z+β⟨E⟩, Helmholtz free energy F(β)=−(1/β)ln Z, and heat capacity C_V. In the thermodynamic limit (large number of sites N) the entropy vanishes as β→±∞, satisfying the third law, while at β=0 (ω=½) the entropy reaches its maximum S_max=ln N. The derivatives of S and F exhibit a discontinuity at β=0, reflecting a sharp change in analytic behavior across the temperature transition.

The nonequilibrium analysis studies the time evolution of the walk’s density matrix ρ