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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 $\omega_{\rm bi}(A)=Z_{\rm bi}^{-1}\sum_n e^{-\beta E_n}\langle\phi_n|A|\psi_n\rangle$ satisfies $\omega_{\rm bi}(A^\dagger A)\ge0$ for all $A$ if and only if $H$ is quasi-Hermitian. The proof reconstructs the metric $\eta$ directly from the eigenprojectors of $\omega_{\rm bi}$ 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 $\omega_\eta(A)=Z_\eta^{-1}{\rm Tr}[\eta e^{-\beta H}A]$ 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 $[\eta,h]\neq0$, 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 $C^*$-norm and the proof of $\sigma$-weak continuity remain open.

Submission history

From: Chen Lan [view email]
[v1] Thu, 11 Jun 2026 12:07:54 UTC (47 KB)
[v2] Tue, 30 Jun 2026 11:10:36 UTC (55 KB)