Interplay between Standard Quantum Detailed Balance and Thermodynamically Consistent Entropy Production
Xin-Hai Tong, Kohei Yoshimura, Tan Van Vu, Naruo Ohga
TL;DR
The paper investigates when Standard Quantum Detailed Balance ($SQDB$) coincides with zero entropy production rate ($EPR$) for quantum Markov semigroups (QMS). It shows two main results: $SQDB$ implies a zero-$EPR$ special representation, and a thermodynamically consistent generator with zero $EPR$ implies $SQDB$, with proofs relying on GKSL representations, special representations, and a symmetric unitary relation $\rho^{1/2} L_k^* = \sum_{\ell} u_{k\ell} L_{\ell} \rho^{1/2}$. Key technical tools include the Autonne–Takagi factorization to normalize the involution matrix $\Delta$ and the structural analysis of the unitary equivalence class of representations. The results clarify the thermodynamic interpretation of quantum dynamics and provide a practical framework to construct zero-$EPR$ representations from $SQDB$ (and conversely) while revealing the limitations via counterexamples.
Abstract
We demonstrate that if a quantum Markovian semigroup satisfies the standard quantum detailed balance condition, its generator admits a special representation that yields a vanishing entropy production rate. Conversely, if the generator admits a special representation adhering to the condition of thermodynamic consistency and leading to a vanishing entropy production rate, then the corresponding quantum Markovian semigroup must satisfy the standard quantum detailed balance condition. In this context, we adopt the definition of entropy production rate that is motivated by the physics literature and standard for thermodynamically consistent Lindbladians.