Non-Hermitian entropy production from fluctuation theorems

How can we rigorously define entropy production and a second law of thermodynamics for non-Hermitian quantum systems? While non-Hermitian dynamics are widely used to model open systems with gain and loss, a consistent thermodynamic interpretation—particularly regarding the role of exceptional points—has remained elusive.

The authors ground their framework in the physics of quantum trajectories. By conditioning the evolution on a specific outcome (the no-jump trajectory), they map non-Hermitian dynamics to a post-selected fluctuation theorem. This allows them to define a stochastic entropy production that remains positive. They further decompose this quantity into incoherent (dissipative) and coherent (non-normal) contributions, linking the latter to information-theoretic measures like Petz-Rényi divergences.

The study establishes that non-Hermitian entropy production is composed of the standard average entropy change plus a "non-Hermitian correction" term. This correction is invariant under Hamiltonian rescaling, distinguishing it from previous definitions that were gauge-dependent. Crucially, the coherent part of this entropy production is non-zero only when the Hermitian and anti-Hermitian parts of the effective Hamiltonian do not commute—a condition necessary for the existence of exceptional points. The authors demonstrate that this coherent component exhibits high sensitivity to parameter variations near exceptional points, providing a thermodynamic signature of these singular features.

This work provides a first-principles thermodynamic foundation for non-Hermitian physics, bridging the gap between abstract non-Hermitian dynamics and physical thermodynamics. By linking entropy production to observable features like exceptional points, the framework offers a new diagnostic tool for experimental platforms such as superconducting circuits and quantum-optical systems, where non-Hermitian effects are increasingly relevant.