Kubo-Martin-Schwinger conditions for non-Hermitian systems

Abstract

We investigate the extension of the Kubo--Martin--Schwinger (KMS) thermal equilibrium condition to bounded non-Hermitian Hamiltonians with real spectra and biorthogonal eigensystems, providing a unified framework through three complementary constructions: a complete KMS theorem under quasi-Hermiticity, a biorthogonal KMS-type identity whose positivity characterises quasi-Hermiticity, and a quantum-detailed-balance condition for the associated open-system dynamics. Our main result is a thermodynamic characterisation of quasi-Hermiticity: for any diagonalisable $H\in M_d(\mathbb{C})$ with real spectrum, the biorthogonal Gibbs functional ${\omega }_{\mathrm{bi}}(A)=Z_{\mathrm{bi}}^{-1}{\sum }_n{e}^{-\beta E_n}⟨{\varphi }_n|A|{\psi }_n⟩$ satisfies ${\omega }_{\mathrm{bi}}(A^{†}A)\ge 0$ for all $A$ if and only if $H$ is quasi-Hermitian. The proof reconstructs the metric $\eta$ directly from the eigenprojectors of ${\omega }_{\mathrm{bi}}$ via the Riesz representation theorem, yielding a metric-free criterion for quasi-Hermiticity. Under the quasi-Hermitian hypothesis, we prove that the $\eta$-Gibbs state ${\omega }_{\eta }(A)=Z_{\eta }^{-1}\mathrm{Tr}[\eta e^{-\beta H}A]$ satisfies the full analytic KMS condition using the Hadamard three-line theorem and Bari's theorem on Riesz bases. The transported state generally differs from the Gibbs state of the isospectral Hermitian partner whenever $[\eta ,h]\ne 0$, so the KMS property cannot be obtained by similarity transformation alone. Finally, within the Haag--Hugenholtz--Winnink programme, we establish the Tomita--Takesaki modular structure of the $\eta$-Gibbs state in finite dimensions, while the construction of a compatible $C^{\ast }$-norm and the proof of $\sigma$-weak continuity remain open.

Major revision with new subsection and expanded references, 47 pages