1 Introduction↩︎

The Kubo–Martin–Schwinger (KMS) condition provides the mathematically rigorous characterisation of thermal equilibrium in quantum statistical mechanics [1], [2]. In the operator-algebraic framework of Haag, Hugenholtz, and Winnink (HHW) [3], a state on a \(C^*\)-algebra is said to be a KMS state at inverse temperature \(\beta > 0\) if its two-point correlation functions admit analytic continuation to a strip in the complex time plane and satisfy a specific boundary condition relating their boundary values. Unlike the Gibbs density matrix formulation, the KMS condition remains meaningful in the thermodynamic limit and has therefore become the fundamental notion of equilibrium in algebraic quantum statistical mechanics [4]. Its structural role has been further illuminated by Tomita–Takesaki modular theory [5] and by its appearance in the Unruh and Hawking effects [6], [7].

In recent decades, non-Hermitian quantum systems have attracted considerable attention across mathematical and theoretical physics [8]–[12]. Prominent examples include \(\mathcal{PT}\)-symmetric quantum mechanics, pseudo-Hermitian and quasi-Hermitian operator theory [13]–[15], and effective descriptions of open quantum systems [16], [17]. Of particular importance is the class of quasi-Hermitian Hamiltonians satisfying \(H^\dagger = \eta H \eta^{-1}\), where \(\eta\) is a bounded positive-definite metric operator. Such Hamiltonians are unitarily equivalent to Hermitian operators and therefore possess real spectra and unitary dynamics in the modified inner product induced by \(\eta\) [13]–[15]. At the same time, effective non-Hermitian Hamiltonians arise naturally in open quantum systems via Lindblad-type dynamics and quantum trajectory methods [11], [16], [17], as well as in condensed matter contexts where non-Hermitian band topology and the skin effect have emerged as central phenomena [12], [18].

These developments raise a fundamental question: to what extent can the notion of thermal equilibrium, as formalised by the KMS condition, be extended beyond the Hermitian setting? This question is not merely formal. The physical interpretation of temperature and equilibrium in non-Hermitian systems—whether arising from \(\mathcal{PT}\)-symmetric quantum mechanics, quasi-Hermitian models, or the effective Hamiltonian of an open quantum system—depends critically on whether a rigorous KMS framework can be established [19]–[22].

A natural starting point is the observation that a diagonalisable non-Hermitian Hamiltonian with a real spectrum admits a biorthogonal eigendecomposition. One may then define thermal expectation values through biorthogonal traces, leading to correlation functions that formally resemble their Hermitian counterparts. However, the KMS condition is substantially stronger than a formal boundary relation. Beyond the boundary identity itself, it requires analyticity of the relevant correlation functions in a complex strip, uniform boundedness, and positivity of the underlying state [4]. While these properties are automatic in the conventional Hermitian framework, they become non-trivial in the non-Hermitian setting and may fail independently. In particular, positivity of the biorthogonal thermal functional is not guaranteed by spectral reality alone, and its failure has direct thermodynamic consequences. Consequently, the existence of a meaningful thermal state for a non-Hermitian system cannot be inferred solely from spectral considerations [21], [23], [24].

The purpose of the present work is to investigate the KMS condition for non-Hermitian quantum systems from a functional-analytic perspective. We consider three closely related settings. First, for quasi-Hermitian Hamiltonians equipped with a positive-definite metric operator \(\eta\), we show that the associated \(\eta\)-Gibbs state satisfies the full KMS condition with respect to the \(\eta\)-modified dynamics. This establishes a rigorous equilibrium framework for a broad class of non-Hermitian systems that are spectrally equivalent to Hermitian ones [13], [15]. Second, we examine the biorthogonal thermal functional constructed directly from the left and right eigenvectors of a diagonalisable Hamiltonian. While this functional generically satisfies the formal KMS boundary relation, we show that positivity is a far more restrictive property. In finite dimensions, positivity of the biorthogonal thermal state is equivalent to quasi-Hermiticity of the Hamiltonian. This result provides a thermodynamic characterisation of quasi-Hermitian systems formulated entirely at the level of thermal states, and complements recent structural results on statistical mechanics for such systems [21]. Third, we discuss open quantum systems governed by Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) dynamics. In this context, the equilibrium properties are determined by the full quantum dynamical semigroup rather than by the effective non-Hermitian Hamiltonian alone [25], [26]. We review the role of quantum detailed balance and clarify its relation to KMS-type stationary states [27], [28].

Our analysis clarifies the precise conditions under which a non-Hermitian system admits a genuine KMS description and identifies the mechanisms responsible for its breakdown, including the loss of positivity, the failure of biorthogonal completeness at exceptional points [12], and the appearance of complex spectra. Taken together, these results establish a unified functional-analytic picture of thermal equilibrium for several important classes of non-Hermitian quantum systems. The paper is organised as follows. Section 2 reviews the standard KMS condition and the non-Hermitian operator framework employed throughout. Sections 3 and 4 develop the quasi-Hermitian and biorthogonal routes respectively, culminating in Theorems 9 and 13. Section 5 treats the open-system setting. Section 6 analyses failure modes at exceptional points and for complex spectra. Section 7 collects the main results and open problems. Appendices 8–10 contain the supporting proofs.

2 KMS condition and non-Hermitian systems↩︎

We begin by reviewing the KMS condition for Hermitian quantum systems, which provides the algebraic characterisation of thermal equilibrium. Then, we introduce the class of non-Hermitian Hamiltonians to be studied throughout this work, and explain why their structural properties make a generalisation of the KMS framework both natural and tractable.

The standard KMS condition↩︎

Let \(H = H^\dagger\) be a self-adjoint Hamiltonian on a Hilbert space \(\mathcal{H}\). At inverse temperature \(\beta > 0\), the system is described by the Gibbs state \[\require{physics} \rho_\beta = \frac{{\rm e}^{-\beta H}}{Z}, \qquad Z := \Tr \left({\rm e}^{-\beta H}\right),\] and observables evolve under the Heisenberg dynamics \[\sigma_t(A) = {\rm e}^{{\rm i}H t} A {\rm e}^{-{\rm i}H t}, \qquad t \in \mathbb{R}.\] The thermal expectation value defines a linear functional \[\require{physics} \omega(X) = \Tr(\rho_\beta X).\] To formulate thermal equilibrium analytically, we introduce an open strip \[\mathcal{S}_\beta = \left\{z \in \mathbb{C} : 0 < \Im z < \beta\right\}\] and its closure \(\overline{\mathcal{S}}_\beta\), obtained by including the two real boundary lines \(\Im z = 0\) and \(\Im z = \beta\).

where \(F_{BA}(z) = \omega\left(\sigma_z(B)\,A\right)\).

We stress that the analyticity requirement is indispensable. The boundary relation Eq. 1 alone, without the strip analyticity and uniform boundedness of \(F_{AB}\) and \(G_{AB}\), is in general insufficient to establish a genuine KMS state. Condition Eq. 1 has a transparent physical meaning: the two thermal correlators \(\omega(A\,\sigma_t(B))\) and \(\omega(\sigma_t(B)\,A)\) arise as boundary values of a single analytic function on \(\mathcal{S}_\beta\), with the imaginary shift \(t \to t + {\rm i}\beta\) encoding the thermal periodicity. In the finite-dimensional setting this relation follows directly from the cyclicity of the trace. In the thermodynamic limit, where a normalised Gibbs density matrix need not exist, the KMS condition survives as the defining characterisation of thermal equilibrium. This is a point of view advocated in the Haag–Hugenholtz–Winnink (HHW) formulation of algebraic quantum statistical mechanics [3], [4].

Non-Hermitian systems↩︎

Non-Hermitian Hamiltonians arise naturally in open quantum systems, effective descriptions, and \(\mathcal{PT}\)-symmetric theories [9], [10], [14], [15], [29]. Among the various generalisations of Hermiticity, pseudo-Hermiticity and its positive-definite specialisation, quasi-Hermiticity, provide the structural foundation needed to extend thermal concepts, including the KMS condition, beyond the conventional self-adjoint framework.

The three structures introduced above are not independent. Quasi-Hermiticity implies a particularly clean interplay between them, as summarised in the following proposition.

The completeness relation Eq. 3 , guaranteed by Prop. 1, is the key ingredient that enables thermal correlation functions and KMS-type identities to be formulated for non-Hermitian systems in direct analogy with the Hermitian case. The remainder of this paper develops this program in detail.

3 Route I: Spectral KMS theorem for quasi-Hermitian systems↩︎

In this section, we work within a functional-analytic framework adapted to quasi-Hermitian Hamiltonians and show that the associated \(\eta\)-Gibbs state admits correlation functions satisfying the three analytic requirements of the KMS condition.

3.1 Assumptions and the \(\eta\)-Hilbert space↩︎

The proofs in this section rest on four conditions, collectively labelled (1–4). The first two encode the algebraic structure of the non-Hermitian system. The third and fourth are analytic regularity conditions required for the KMS analyticity argument.

The two-sided bound on \(\eta\) ensures that the \(\eta\)-weighted inner product \(\langle \cdot | \eta | \cdot \rangle\) is equivalent to the standard inner product on \(\mathcal{H}\), so that the \(\eta\)-Hilbert space \(\mathcal{H}_\eta\) and \(\mathcal{H}\) carry the same topological structure.

Physical Hamiltonians, e.g. harmonic oscillators, Schrödinger operators, lattice Laplacians, are typically unbounded self-adjoint operators on \(\mathcal{H}\). The assumption \(H \in \mathcal{B}(\mathcal{H})\) is a mathematical simplification: it makes the operator exponential \({\rm e}^{{\rm i}H t} = \sum_{n=0}^\infty ({\rm i}t)^n H^n / n!\) norm-convergent and validates the interchange of sums and bounded operators throughout the proofs below. For physically relevant unbounded \(H\), the exponential \({\rm e}^{{\rm i}Ht}\) is defined via Stone’s theorem or the spectral functional calculus, and the KMS framework can be extended using domain-theoretic methods (cf. [4]). Such an extension is deferred to future work.

This is the defining relation of pseudo-Hermiticity with metric \(\eta\). Combined with the positive-definiteness of \(\eta\) in Assum. 1, it promotes \(H\) to a quasi-Hermitian operator (Def. 3) and guarantees, in particular, that the spectrum of \(H\) is real (Prop. 1).

Two additional spectral finiteness conditions are imposed independently, as neither implies the other:

• (Spectral lower bound.) \(E_{\min} := \inf_n E_n > -\infty\). This prevents the operator norm \(\|{\rm e}^{{\rm i}Hz}\|\) from diverging on the closed strip \(\overline{\mathcal{S}}_\beta\): without it, terms \({\rm e}^{-\alpha E_n}\) grow without bound as \(E_n \to -\infty\), invalidating the uniform estimates in Prop. 7.

• (Partition function finiteness.) \(Z_\eta := \sum_n {\rm e}^{-\beta E_n} < \infty\). Note that \(E_{\min} > -\infty\) does not imply \(Z_\eta < \infty\): for instance, \(E_n = \ln n\) gives \(Z_\eta = \sum n^{-\beta}\), which diverges for \(\beta \leq 1\). Both conditions must therefore be imposed separately.

The Riesz basis condition, combined with the real spectrum and the spectral lower bound, ensures that the operator exponential \({\rm e}^{{\rm i}Hs} = \sum_j {\rm e}^{{\rm i}E_j s} |\psi_j\rangle\langle\phi_j|\) converges in operator norm for \(s\) in any bounded subset of \(\mathbb{C}\). This follows from the Bari theorem [30]: for a Riesz basis \(\{|\psi_j\rangle\}\) with biorthogonal dual \(\{|\phi_j\rangle\}\), the series \(\sum_j c_j |\psi_j\rangle\langle\phi_j|\) converges in operator norm whenever \(\{c_j\} \in \ell^\infty\). Since \(E_j \in \mathbb{R}\), the coefficients \({\rm e}^{{\rm i}E_j s}\) are bounded for \(s\) in any strip of finite width, and the conclusion follows.

The Riesz basis property is in fact a consequence of Assum. 1: the map \(U = \eta^{1/2}\), which is bounded and boundedly invertible by Assum. 1, transforms the \(\eta\)-orthonormal basis \(\{|\psi_n\rangle\}\) into the standard orthonormal basis \(\{U|\psi_n\rangle\}\). Hence \(|\psi_n\rangle = U^{-1}(U|\psi_n\rangle)\) is the image of a standard ONB under the bounded invertible map \(U^{-1}\), which is the defining property of a Riesz basis.

This condition guarantees that the double series appearing in the proof of Prop. 6 converge absolutely on \(\partial\overline{\mathcal{S}}_\beta\), which is the key step in verifying strip analyticity.

The bi-HS condition is tied to the choice of biorthogonal system \(\{|\psi_n\rangle, |\phi_n\rangle\}\) and is not an intrinsic, basis-independent property of the operator \(X\). It is used here purely as a sufficient summability condition and should not be interpreted as membership in an operator ideal. In finite dimensions, every operator is bi-HS with respect to any basis, so the condition is automatically satisfied.

The following example identifies two physically natural classes of observables satisfying Assum. 4 in the infinite-dimensional setting, confirming that the assumption is not restrictive in practice.

We now construct the Hilbert space and operator framework within which the KMS proof will be carried out. The key idea is to intertwine \(H\) with a genuine self-adjoint operator \(h\) via the bounded invertible map \(U := \eta^{1/2}\), and to transfer the entire thermal analysis to \(h\).

Define \[h := U H U^{-1} = \eta^{1/2} H \eta^{-1/2}. \label{eq:h-def}\tag{4}\]

Lemma 1 confirms that all results from Hermitian spectral theory and operator semigroups apply to \(h\), and can be transferred back to \(H\) via the bounded intertwining map \(U\).

We equip \(\mathcal{H}\) with the \(\eta\)-inner product \[\langle\varphi, \psi\rangle_\eta := \langle\varphi | \eta | \psi\rangle,\] and write \(\mathcal{H}_\eta\) for the resulting Hilbert space. The \(\eta\)-adjoint of an operator \(A\) is defined by \[A^{\dagger_\eta} := \eta^{-1} A^\dagger \eta, \label{eq:eta-adj}\tag{5}\] and a direct computation using Assum. 2 gives \[H^{\dagger_\eta} = \eta^{-1} H^\dagger \eta = \eta^{-1}(\eta H \eta^{-1})\eta = H,\] confirming that \(H\) is self-adjoint in \(\mathcal{H}_\eta\), as expected from the quasi-Hermitian structure.

Two further identities follow from \(\eta = U^2\) and the intertwining relation \({\rm e}^{-\beta H} = U^{-1} {\rm e}^{-\beta h} U\): \[\require{physics} \begin{equation} \eta\, {\rm e}^{-\beta H} = U\, {\rm e}^{-\beta h}\, U, \tag{6} \end{equation} \begin{equation} Z_\eta:= \Tr[\eta\, {\rm e}^{-\beta H}] = \Tr[{\rm e}^{-\beta h}\,\eta] = \sum_n {\rm e}^{-\beta E_n}, \tag{7} \end{equation}\] where the second equality in Eq. 7 uses the cyclicity of the trace and \(\eta = U^2\), and the last equality uses the diagonalisability Assum. 3. By Assum. 3, \(Z_\eta < \infty\), so the partition function is well defined.

With these preparations, the \(\eta\)-Gibbs state is \[\require{physics} \rho_\beta^{(\eta)}:= \frac{{\rm e}^{-\beta H}}{Z_\eta}, \qquad \omega_\eta(A):= \Tr_\eta \left[\rho_\beta^{(\eta)}\, A\right] = \frac{\Tr[\eta\, {\rm e}^{-\beta H} A]}{Z_\eta}. \label{eq:eta-gibbs}\tag{8}\] This is the non-Hermitian analogue of the Gibbs state: the factor \(\eta\) in the numerator weights the trace by the metric, ensuring that \(\omega_\eta\) is a positive functional on \(\mathcal{H}_\eta\).

3.2 Time evolution and \(\eta\)-Gibbs state↩︎

Before establishing the KMS condition, we verify that the Heisenberg time evolution \(\sigma_t\) is compatible with the \(\eta\)-adjoint structure. This compatibility (the \(*\)-automorphism property ) is a prerequisite for the thermal functional \(\omega_\eta\) to define a genuine state in the \(\eta\)-algebraic sense. Without it, the notion of “\(\eta\)-positive” and the KMS boundary condition would not be internally consistent.

Having established that \(\sigma_t\) respects the \(\eta\)-adjoint, we now show that the \(\eta\)-Gibbs state \(\omega_\eta\) defined in Eq. 8 is positive and faithful with respect to the same structure. Positivity is indispensable for a physical interpretation: it guarantees that \(\omega_\eta(A^{\dagger_\eta} A)\) is a non-negative real number for every observable \(A\), playing the role that positivity of the standard inner product plays in ordinary quantum mechanics. Faithfulness — strict positivity for \(A \neq 0\) — is equally important, and it ensures that no non-zero observable is invisible to the state, which is required for the KMS modular theory to be non-degenerate.

3.3 Main theorem: The spectral KMS condition↩︎

We now assemble the preceding results into a proof of the main theorem. The argument proceeds in three stages. First, we derive a spectral matrix-element identity (Lem. 2) that encodes the action of the complex-time evolution on biorthogonal matrix elements. Second, we use this identity together with the bi-HS condition Assum. 4 to establish analyticity and boundedness of the thermal correlation function on the closed strip \(\overline{\mathcal{S}}_\beta\) (Prop. 6). Third, we verify that the correlation function coincides with the \(\eta\)-state functional \(\omega_\eta(A\,\sigma_z(B))\) everywhere on \(\overline{\mathcal{S}}_\beta\) (Prop. 7), completing the verification of all three conditions of Def. 1.

Stage 1: Spectral matrix-element identity↩︎

Stage 2: Analyticity and boundedness of the correlation function↩︎

With Lem. 2 in hand, we examine the analytic properties of the thermal two-point function \[G_{AB}(z) := \frac{1}{Z_\eta} \sum_{n,m} {\rm e}^{-\beta E_n} {\rm e}^{{\rm i}(E_m - E_n)z} A_{nm} B_{mn}, \label{eq:G-def}\tag{14}\] which arises naturally from expanding \(\omega_\eta(A\,\sigma_z(B))\) in the biorthogonal basis.

Stage 3: Identification with the thermal functional↩︎

It remains to show that the series \(G_{AB}(z)\) defined in Eq. 14 coincides with the \(\eta\)-state functional \(\omega_\eta(A\,\sigma_z(B))\) throughout \(\overline{\mathcal{S}}_\beta\), thereby linking the analytic function constructed in Stage 2 to the physical thermal correlator.

The spectral KMS theorem↩︎

3.4 Frequency-domain formulation and relation to the HHW theorem↩︎

We close this section with two complementary perspectives on Thm. 9. The first reformulates the KMS condition in the frequency domain, yielding a generalised detailed-balance relation that makes the physical content of thermal equilibrium transparent. The second situates Thm. 9 within the broader framework of algebraic quantum statistical mechanics, clarifying both what has been established relative to the Bratteli–Robinson (BR) formulation and why the result is not a trivial consequence of the standard Hermitian theory.

Frequency-domain KMS condition↩︎

The biorthogonal spectral function associated with the pair \((A, B)\) is defined by \[\require{upgreek} \rho_{AB}(\omega) := \frac{2\uppi}{Z_\eta} \sum_{n,m} {\rm e}^{-\beta E_n} A_{nm} B_{mn}\, \delta \left(\omega - (E_n - E_m)\right). \label{eq:spectral-fn}\tag{20}\] This is the non-Hermitian analogue of the spectral density familiar from the fluctuation-dissipation theorem: its support is concentrated on the Bohr frequencies \(E_n - E_m\) of the system, and its weights are governed by the thermal factor \({\rm e}^{-\beta E_n}\) together with the biorthogonal matrix elements \(A_{nm}\), \(B_{mn}\).

The relation Eq. 21 is a direct generalisation of the standard fluctuation-dissipation theorem to quasi-Hermitian systems: the factor \({\rm e}^{\beta\omega}\) encodes the asymmetry between emission and absorption at frequency \(\omega\), and its form is identical to the Hermitian case, confirming that thermal equilibrium in the \(\eta\)-inner product space retains the characteristic signature of the KMS condition.

Precise status relative to the HHW theorem↩︎

Theorem 9 establishes the KMS property at the level of bounded operators on a Hilbert space equipped with a positive-definite metric \(\eta\). To contextualise this result within the BR formulation of algebraic quantum statistical mechanics [4], we record in Tab. 1 which of the BR-KMS requirements have been verified and which remain open.

The three open items in Tab. 1 are structural rather than analytic: they require equipping the observable algebra with a \(C^*\)-norm [31], establishing strong continuity of \(\sigma_t\) in the \(\sigma\)-weak operator topology, and constructing the Tomita–Takesaki modular theory for the \(\eta\)-Gibbs state. These constitute a natural program for future work. The analytic core of the KMS condition — the content that is directly tied to physical predictions such as the fluctuation-dissipation theorem — has been fully verified.

Why the result is not a consequence of the standard Hermitian theory↩︎

Since \(h = UHU^{-1}\) is self-adjoint (Lem. 1), one might ask whether Thm. 9 is simply the standard Hermitian KMS theorem transported through the algebra isomorphism \(\Phi:= A \to UAU^{-1}\) induced by \(U = \eta^{1/2}\). The answer is no, and understanding precisely why clarifies the independent contribution of the theorem.

The map \(\Phi\) intertwines the two evolution groups, \(\Phi \circ \sigma_t^H = \sigma_t^h \circ \Phi\), so the dynamical systems \((\mathcal{B}(\mathcal{H}), \sigma_t^H, \omega_\eta)\) and \((\mathcal{B}(\mathcal{H}), \sigma_t^h, \hat{\omega})\), where \(\hat{\omega} := \omega_\eta \circ \Phi^{-1}\), are algebraically isomorphic. The KMS property is preserved by algebra isomorphisms, so the two systems are KMS-equivalent. This, however, is a logical equivalence — not a deduction: it says that \(\omega_\eta\) is KMS for \((H, \sigma_t^H)\) if and only if \(\hat{\omega}\) is KMS for \((h, \sigma_t^h)\). To invoke the standard Hermitian theorem on the right-hand side, one would need \(\hat{\omega}\) to be the standard Gibbs state \(\omega_h\) of \(h\). But this is false whenever \([\eta, h] \neq 0\).

To see this explicitly, the transported state is \[\require{physics} \hat{\omega}(X) := \omega_\eta(U^{-1}XU) = \frac{\Tr\left[{\rm e}^{-\beta h} X \eta\right]}{Z_\eta}, \label{eq:hat-omega}\tag{22}\] whereas the standard Gibbs state of \(h\) is \(\require{physics} \omega_h(X) = \Tr[{\rm e}^{-\beta h} X]/Z_h\). The two functionals agree on every \(X\) if and only if \(\require{physics} \Tr[{\rm e}^{-\beta h}[X, \eta]] = 0\) for all \(X\), which holds if and only if \([\eta, h] = 0\). Moreover, their partition functions satisfy \(\require{physics} Z_\eta = \Tr[{\rm e}^{-\beta h}\eta] \neq \Tr[{\rm e}^{-\beta h}] = Z_h\) in general, so the normalisation itself differs. The state \(\hat{\omega}\) is therefore a faithful normal state of the Hermitian system \((h, \sigma_t^h)\) distinct from \(\omega_h\), and its KMS property does not follow from the uniqueness of the Gibbs state as the tracial KMS state of a finite-dimensional Hermitian system.

In summary, the logical structure is as follows. The standard theorem gives the KMS property of \(\omega_h\) for \((h, \sigma_t^h)\). Theorem 9 gives the KMS property of \(\omega_\eta\) for \((H, \sigma_t^H)\). The isomorphism \(\Phi\) shows these two statements are equivalent via \(\hat{\omega} \neq \omega_h\), but neither statement implies the other through existing results. The proof in §3.3 provides a direct, self-contained verification of the KMS analytic conditions for \(\omega_\eta\), using the biorthogonal spectral expansion and the analytic infrastructure of Props 6–7 — objects specific to the non-Hermitian setting that have no counterpart in the Mostafazadeh–Scholtz theory.

Finally, we note that Route II (Sec. 4) lies entirely outside the similarity framework: without a positive-definite \(\eta\), the isomorphism \(\Phi\) is not available, and the biorthogonal KMS-type identity (Prop. 11) and the structure theorem (Thm. 13) are genuinely new results with no counterpart in the Hermitian theory.

4 Route II: Biorthogonal KMS-type identity↩︎

Route I established that \(\eta\)-Gibbs states satisfy the full analytic KMS condition, but it does so under the strong hypothesis of quasi-Hermiticity: a positive-definite metric \(\eta\) is assigned to the system from the outset, and the entire proof machinery — the \(\eta\)-inner product, the intertwining map \(U\), the \(*\)-automorphism property — depends on this structure.

The present section asks a more primitive question: how much of the KMS framework survives if one discards the metric entirely and retains only the two intrinsic spectral properties of a non-Hermitian Hamiltonian, namely reality of eigenvalues and biorthogonal completeness[32]? Working in this leaner setting we construct the biorthogonal Gibbs state \(\omega_{\rm{bi}}\), prove that it satisfies a formal KMS-type identity, and then diagnose precisely where and why the full KMS condition fails without \(\eta\).

4.1 Assumptions and the biorthogonal trace↩︎

Route II operates under the following two standing assumptions, which are strictly weaker than (A1–A4).

By Prop. 1, (A1–A4) implies (A5)+(A6), but the converse fails: a non-Hermitian operator may have real spectrum and a complete biorthogonal basis without admitting any positive-definite intertwining metric.

The natural trace functional in this setting is the biorthogonal trace \[\require{physics} \Tr_{\rm{bi}}[A] := \sum_n \langle\phi_n | A | \psi_n\rangle = \sum_n A_{nn}, \label{eq:bi-trace}\tag{23}\] where the sum converges absolutely under (A3) for the operators of interest. Unlike the standard operator trace, \(\require{physics} \Tr_{\rm{bi}}\) depends on the biorthogonal system \(\{|\psi_n\rangle, |\phi_n\rangle\}\) and is not unitarily invariant. However, it satisfies a cyclicity property that is the direct analogue of the cyclicity of the standard trace and plays the same structural role in the proofs below.

4.2 The biorthogonal Gibbs state and the KMS-type identity↩︎

With the biorthogonal trace in hand, we define the thermal state associated with the non-Hermitian Hamiltonian under Route II hypotheses, and establish the central identity of this section.

The biorthogonal Gibbs state↩︎

The biorthogonal Gibbs state is the density operator \[\rho_\beta^{\rm{bi}} := \frac{1}{Z_{\rm{bi}}} \sum_n {\rm e}^{-\beta E_n} |\psi_n\rangle\langle\phi_n|, \label{eq:bi-density}\tag{24}\] whose associated functional is \[\require{physics} \omega_{\rm{bi}}(A) := \Tr_{\rm{bi}} \left[\rho_\beta^{\rm{bi}}\, A\right] = \frac{1}{Z_{\rm{bi}}} \sum_n {\rm e}^{-\beta E_n} A_{nn}. \label{eq:bi-gibbs}\tag{25}\] The structure of \(\rho_\beta^{\rm{bi}}\) mirrors the standard Gibbs density matrix, with the orthonormal projectors \(|\psi_n\rangle\langle\psi_n|\) replaced by the biorthogonal projectors \(|\psi_n\rangle\langle\phi_n|\).

Two basic properties are immediate. First, \(\omega_{\rm{bi}}\) is time-translation invariant: by Lem. 2 applied with \(m = n\), \((A(t))_{nn} = {\rm e}^{{\rm i}(E_n - E_n)t} A_{nn} = A_{nn}\), so \(\omega_{\rm{bi}}(\sigma_t(A)) = \omega_{\rm{bi}}(A)\) for all \(t \in \mathbb{R}\). Second, \(\omega_{\rm{bi}}\) is normalised: \(\omega_{\rm{bi}}(\mathbf{1}) = Z_{\rm{bi}}^{-1} \sum_n {\rm e}^{-\beta E_n} = 1\).

A crucial caveat is that \(\omega_{\rm{bi}}\) is not generally positive. The standard positivity condition \(\omega_{\rm{bi}}(A^\dagger A) \geq 0\) fails because the inner product \(\langle\phi_n | A^\dagger | \psi_k\rangle = \overline{\langle\psi_k | A | \phi_n\rangle}\) involves \(|\phi_n\rangle\) in the ket position, which differs from \(|\psi_n\rangle\) unless \(|\phi_n\rangle \propto \eta|\psi_n\rangle\) — a condition that is precisely quasi-Hermiticity. Positivity is thus generically absent in Route II, and its restoration is the content of Thm. 13.

The KMS-type identity↩︎

4.3 The gap between Route II and full KMS↩︎

Proposition 11 shows that the biorthogonal state \(\omega_{\rm{bi}}\) satisfies the time-domain boundary relation and the strip analyticity of the KMS condition. What it does not do is establish a genuine KMS state in the sense of Def. 1, because Def. 1 implicitly requires the underlying functional to be a state — that is, a positive normalised functional. The gap between Route II and full KMS is therefore not analytic but algebraic: it lies precisely in the failure of positivity.

Table 2 records the precise status of each KMS-relevant condition under Route II.

The dividing line runs through the middle of the table. The upper block — the analytic and algebraic requirements that are directly tied to the time-domain boundary relation — holds under Route II alone. The lower block — positivity, the \(*\)-automorphism property in the standard inner product, and the \(C^*\)-algebraic GNS construction — all require the additional structure of Route I.

The obstruction is not merely a technical inconvenience. Positivity is the condition that makes \(\omega_{\rm{bi}}\) a physical state: without it, the functional assigns negative “probabilities” to some observables and loses its interpretation as a thermal ensemble. The GNS construction, which builds the Hilbert space representation of the \(C^*\)-algebra from the state, requires positivity as an input. Without it, the inner product on the GNS space is indefinite and the construction breaks down. The \(*\)-automorphism property of \(\sigma_t\) in the standard inner product (Thm. 2) likewise depends on the pseudo-Hermitian relation \(H^\dagger = \eta H\eta^{-1}\), which is Route I data.

Theorem 13 makes this gap precise: it characterises quasi-Hermiticity as the exact condition under which \(\omega_{\rm{bi}}\) is positive, thereby showing that Route I is not merely sufficient but also necessary for \(\omega_{\rm{bi}}\) to be a genuine KMS state. In this sense, the biorthogonal KMS-type identity ?? is best understood not as a weakened version of the full KMS condition, but as a diagnostic: it holds universally under Route II, and its upgrade to a full KMS state is equivalent to the system being quasi-Hermitian.

4.4 Structure theorem: KMS positivity characterises quasi-Hermiticity↩︎

The gap analysis in §4.3 identified positivity of \(\omega_{\rm{bi}}\) as the sole condition separating the biorthogonal KMS-type identity from a genuine KMS state. The theorem of this subsection closes that gap from both directions: it shows that \(\omega_{\rm{bi}}\) is positive if and only if \(H\) is quasi-Hermitian, thereby providing an intrinsic, metric-free characterisation of quasi-Hermiticity in terms of the thermal functional alone.

This is the central new result of the paper. Unlike the Mostafazadeh–Scholtz framework, which starts from a given metric \(\eta\) and studies its consequences, the theorem starts from a property of the state — positivity of \(\omega_{\rm{bi}}\) — and recovers the metric as a derived object. The implication (i)\(\Rightarrow\)(ii) is therefore strictly outside any similarity-transformation paradigm.

The proof is structured as follows. The implications (ii)\(\Rightarrow\)(iii)\(\Rightarrow\)(i) are straightforward: quasi-Hermiticity determines the left eigenvectors up to a common metric, and once \(|\phi_n\rangle = \eta|\psi_n\rangle\) is established, positivity is a one-line computation. The substantial direction is (i)\(\Rightarrow\)(ii): given only that \(\omega_{\rm{bi}}\) is positive, we must construct a positive-definite \(\eta\) satisfying \(H^\dagger = \eta H\eta^{-1}\). The key is to define \(\eta_0 := \sum_n |\phi_n\rangle\langle\phi_n|\), show it is positive-definite (Lem. 4), and verify the pseudo-Hermitian relation directly from the spectral decompositions. The positivity hypothesis enters through the Riesz representation theorem, which forces the representing density matrix of \(\omega_{\rm{bi}}\) to be positive-semidefinite, and this in turn constrains the geometry of the biorthogonal system.

The proof of the theorem admits a transparent matrix-algebraic interpretation. The Gram-type matrix \(G = \Phi D_\beta \Phi^{-1}\) arising from the Riesz representation is diagonalised by the left eigenvectors \(|\phi_n\rangle\) with thermal eigenvalues \({\rm e}^{-\beta E_n}\). Positivity of \(G\) — enforced by the thermal weights being strictly positive — is what makes \(\{|\phi_n\rangle\}\) a genuine basis (not just a formal biorthogonal set), which is precisely the input needed for Lem. 4 and the construction Eq. 31 to succeed.

5 Route III: Lindblad quantum detailed balance↩︎

Routes I and II operate within the framework of closed quantum systems: the Hamiltonian \(H\) is a fixed operator on a Hilbert space, and thermal equilibrium is characterised by the KMS condition on the time-evolved correlators. When the non-Hermitian Hamiltonian arises instead as the effective description of an open quantum system — one that exchanges energy or particles with an environment — the appropriate framework is the full Lindblad master equation, and the notion of thermal equilibrium is replaced by that of a steady state satisfying quantum detailed balance (QDB).

This section develops Route III: we set up the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) formalism, clarify the relationship between QDB and the KMS condition in this setting, and state the Fagnola–Umanità characterisation theorem for Davies generators. The section closes by locating Route III within the broader framework of the paper and explaining why it is logically independent of Routes I and II.

5.1 GKSL equation and the effective Hamiltonian↩︎

The GKSL master equation governing the time evolution of the density matrix \(\rho(t)\) of an open quantum system is \[\partial_t \rho = \mathcal{L}[\rho] := -i[H_{\rm{sys}}, \rho] + \sum_k \left( L_k \rho L_k^\dagger - \tfrac{1}{2}\{L_k^\dagger L_k, \rho\} \right), \label{eq:lindblad}\tag{32}\] where \(H_{\rm{sys}} = H_{\rm{sys}}^\dagger\) is the self-adjoint system Hamiltonian and \(\{L_k\}\) are the jump operators encoding the coupling to the environment. The Lindbladian \(\mathcal{L}\) generates a completely positive trace-preserving (CPTP) semigroup \(\{{\rm e}^{t\mathcal{L}}\}_{t \geq 0}\) on the space of density matrices, and a steady state \(\rho_{\rm{ss}}\) is defined by the condition \(\mathcal{L}[\rho_{\rm{ss}}] = 0\).

In the no-jump approximation, the dynamics between quantum jumps is governed by the effective non-Hermitian Hamiltonian \[H_{\rm{eff}} = H_{\rm{sys}} - \frac{{\rm i}}{2}\sum_k L_k^\dagger L_k, \label{eq:Heff}\tag{33}\] which is precisely the type of operator studied in Routes I and II. However, \(H_{\rm{eff}}\) captures only the coherent part of the open-system dynamics: the KMS and detailed balance properties of the steady state depend on the full Lindblad data \(\{H_{\rm{sys}}, L_k\}\), not on \(H_{\rm{eff}}\) alone. In particular, two different sets of jump operators \(\{L_k\}\) can yield the same \(H_{\rm{eff}}\) while producing steady states with entirely different thermal properties. This is the fundamental reason why Route III must be developed independently, rather than deduced from Routes I or II by substituting \(H_{\rm{eff}}\) for \(H\).

5.2 Quantum detailed balance↩︎

The notion of thermal equilibrium for open quantum systems is captured by the quantum detailed balance condition, which generalises the classical detailed balance condition \(\rho_i W_{ij} = \rho_j W_{ji}\) (where \(W_{ij}\) are transition rates) to the non-commutative setting. Several inequivalent formulations have been proposed in the literature, and we begin by recording the principal variants and then specialise to the one used throughout this section.

The precise logical relationship between QDB and the KMS condition is as follows. Standard QDB implies that the steady-state correlation functions satisfy a KMS-type relation, but the converse does not hold in general. The correct statement is:

In particular, the claim “KMS is equivalent to quantum detailed balance” — sometimes encountered in the open-systems literature — is too strong. QDB is a sufficient condition for a KMS-type thermal steady state in the Lindblad setting, and one that is tractable to verify through the algebraic condition Eq. 34 , but it carries additional structural content beyond the KMS boundary relation alone.

5.3 Fagnola–Umanità characterisation for Davies generators↩︎

The most natural class of Lindblad generators for which the QDB condition admits a clean algebraic characterisation is that of Davies generators, arising from the Davies weak-coupling limit [34]. In this class, the jump operators are eigenoperators of the \(H_{\rm{sys}}\)-modular automorphism, and the detailed balance condition reduces to a transparent commutation relation between each \(L_k\) and the thermal density matrix.

5.4 Relation to Routes I and II↩︎

Route III is logically independent of Routes I and II, and the relationship between them deserves careful statement.

The most direct link is through the effective Hamiltonian \(H_{\rm{eff}}\) in Eq. 33 : Routes I and II study the KMS properties of a non-Hermitian operator \(H\) that could, in principle, arise as \(H_{\rm{eff}}\) for some choice of \(\{L_k\}\). However, as noted in §5.1, the thermal properties of the open system are determined by the full Lindblad data, not by \(H_{\rm{eff}}\) alone. The KMS condition satisfied by the \(\eta\)-Gibbs state \(\omega_\eta\) in Route I is a property of the closed dynamics generated by \(H\). It does not imply, and is not implied by, the QDB condition Eq. 34 for any particular Lindblad embedding.

More precisely, Routes I/II and Route III occupy different levels of the quantum dynamics hierarchy:

• Routes I and II concern the unitary (or quasi-unitary) dynamics \(\sigma_t(A) = {\rm e}^{iHt}Ae^{-iHt}\) of a closed system with a fixed non-Hermitian Hamiltonian, and characterise thermal equilibrium through the KMS boundary condition on correlation functions.

• Route III concerns the dissipative dynamics \({\rm e}^{t\mathcal{L}}\) of an open system, and characterises thermal equilibrium through the QDB condition on the generator \(\mathcal{L}\). The steady state \(\rho_{\rm{ss}}\) is not a Gibbs state of \(H_{\rm{eff}}\) in general, but a fixed point of the full CPTP semigroup.

The correct summary is therefore: standard QDB is a sufficient condition for a KMS-type thermal steady state in open systems, and this condition is formulated entirely in terms of the Lindblad generator, independently of whether the effective Hamiltonian \(H_{\rm{eff}}\) satisfies the quasi-Hermitian structure of Route I. The three routes are complementary rather than competing characterisations of non-Hermitian thermal equilibrium, each applicable to a different physical regime.

6 Exceptional points and complex spectra↩︎

The positive results of Secs 3 and 4 rest on two spectral hypotheses: reality of the eigenvalues and biorthogonal completeness. This section examines what happens when these hypotheses fail. The two principal failure modes — exceptional points, where diagonalisability breaks down, and complex spectra, where eigenvalues acquire non-zero imaginary parts — are physically distinct and destroy the KMS framework through different mechanisms. Identifying these mechanisms precisely is not merely a matter of logical completeness: it delimits the exact boundary of applicability of the quasi-Hermitian thermal framework and points to the structural features — Jordan-block corrections, complex Boltzmann weights, unbounded correlation functions — that any future extension of the theory must address.

Throughout this section we work at the level of formal spectral series, without assuming (A1–A4) or (A5)+(A6), to isolate where each argument breaks down.

6.1 Structure collapse at exceptional points↩︎

An exceptional point (EP) [35] of order \(m\) is a value \(E_{\rm{EP}}\) at which \(m\) eigenvalues and their corresponding eigenstates simultaneously coalesce. At such a point the Hamiltonian is no longer diagonalisable: it can only be brought to a Jordan normal form. Concretely, the defective eigenvector satisfies \[H|\psi_{\rm{EP}}\rangle = E_{\rm{EP}}|\psi_{\rm{EP}}\rangle, \qquad \text{but} \qquad \langle\phi_{\rm{EP}}|\psi_{\rm{EP}}\rangle = 0, \label{eq:EP-def}\tag{37}\] where the vanishing of the biorthogonal inner product signals the collapse of the dual eigenbasis. The correct replacement for the resolution of the identity is the Jordan chain: vectors \(|\psi^{(0)}\rangle, |\psi^{(1)}\rangle, \ldots, |\psi^{(m-1)}\rangle\) satisfying the recursion \[(H - E_{\rm{EP}})|\psi^{(k)}\rangle = |\psi^{(k-1)}\rangle, \qquad k = 1, \ldots, m-1, \label{eq:jordan-chain}\tag{38}\] with \(|\psi^{(0)}\rangle\) the eigenvector. These chains, together with the non-defective eigenvectors, yield the modified resolution of the identity \[\sum_{k=0}^{m-1} |\psi^{(k)}\rangle\langle\phi^{(m-1-k)}| + (\text{non-defective terms}) = \mathbf{1}. \label{eq:jordan-completeness}\tag{39}\] The Jordan structure has an immediate dynamical consequence. Exponentiating \(H\) via the Jordan decomposition gives \[{\rm e}^{{\rm i}Ht}|\psi^{(0)}\rangle = {\rm e}^{{\rm i}E_{\rm{EP}}t} \sum_{k=0}^{m-1} \frac{({\rm i}t)^k}{k!}\,|\psi^{(k)}\rangle, \label{eq:poly-growth}\tag{40}\] so the time evolution of the defective mode exhibits polynomial growth \(\sim t^{m-1}\) in addition to the oscillatory factor \({\rm e}^{{\rm i}E_{\rm{EP}}t}\). This is the signature of a non-diagonalisable operator and the source of all KMS failures listed below.

Each of the three KMS conditions in Def. 1 is violated at an EP, through a distinct mechanism.

1. Biorthogonal completeness fails, invalidating the spectral expansion. Lemma 2 rests on the identity \(\sum_k |\psi_k\rangle\langle\phi_k| = \mathbf{1}\) from (A6). At an EP this identity is replaced by Eq. 39 , which contains off-diagonal Jordan projectors \(|\psi^{(k)}\rangle\langle\phi^{(m-1-k)}|\) for \(k \geq 1\). Inserting the Jordan resolution into the spectral-element calculation of Lem. 2 produces polynomial corrections: matrix elements contain terms of the form \(s^k {\rm e}^{{\rm i}E_{\rm{EP}} s}\) for \(k = 1, \ldots, m-1\), rather than the purely exponential form \({\rm e}^{{\rm i}(E_m - E_n)s}\) in Eq. 13 . Consequently, the thermal two-point function \(G_{AB}(z)\) acquires terms of the form \(z^k {\rm e}^{{\rm i}\lambda z}\) whose growth properties are qualitatively different from those of pure exponentials.

2. Boundedness on \(\overline{\mathcal{S}}_\beta\) fails. On the real boundary \(\Im(z) = 0\), the polynomial factor in 40 gives \(\|{\rm e}^{{\rm i}Ht}\| \sim |t|^{m-1}\) as \(t \to \pm\infty\). The correlation function \(G_{AB}(t)\) therefore grows polynomially on the real axis, violating KMS condition (ii) (boundedness on \(\overline{\mathcal{S}}_\beta\)). Note that polynomial growth does not destroy complex analyticity — functions of the form \(z^k {\rm e}^{{\rm i}\lambda z}\) are entire — so the analyticity condition (i) may still hold formally. It is the \(L^\infty\) bound on the boundary lines that fails and that the Hadamard three-line theorem requires.

3. The equilibrium automorphism group is ill-defined. The KMS framework, and in particular the Tomita–Takesaki modular theory, requires the time evolution \(\sigma_t\) to extend to a well-posed one-parameter automorphism group on the observable algebra. The polynomial prefactors \(t^k\) in Eq. 40 are incompatible with this requirement: the map \(t \to \sigma_t(A)\) is no longer bounded uniformly in \(t\) for defective modes, and the modular flow \(\Delta^{it}\) of Tomita–Takesaki theory cannot be established.

4. Cyclicity of \(\require{physics} \Tr_{\rm{bi}}\) fails. Lemma 3 uses only the completeness relation \(\sum_n |\psi_n\rangle\langle\phi_n| = \mathbf{1}\), but at an EP this is replaced by Eq. 39 , which is not cyclic under the standard trace argument. Consequently, the biorthogonal KMS-type identity of Prop. 11 also breaks down at exceptional points.

The four failure modes above are not independent: they all trace back to the single structural defect of non-diagonalisability. Once the Jordan chain replaces the eigenbasis, the purely exponential time evolution that underlies every step of the KMS proof — the spectral expansion, the strip bounds, the boundary identity — is corrupted.

A complete KMS framework at exceptional points — encompassing an appropriate notion of equilibrium state, a Jordan-block generalisation of the modular flow, and the correct thermodynamic interpretation of defective modes — remains an open problem.

6.2 Complex Spectrum↩︎

We now turn to a different failure mode: non-Hermitian Hamiltonians with genuinely complex eigenvalues \(E_n = \alpha_n + {\rm i}\gamma_n\) with \(\gamma_n \neq 0\). Unlike exceptional points, where the geometry of the eigensystem collapses, a complex spectrum destroys the KMS framework through the behaviour of the Boltzmann weights and the growth of the correlation function on the real time axis.

When \(\gamma_n \neq 0\), the Boltzmann weight becomes complex: \[{\rm e}^{-\beta E_n} = {\rm e}^{-\beta\alpha_n} {\rm e}^{-{\rm i}\beta\gamma_n} \in \mathbb{C}, \label{eq:complex-weight}\tag{41}\] so the partition function \(Z = \sum_n {\rm e}^{-\beta E_n}\) is no longer real and positive, and loses its interpretation as a normalisation constant for a probability distribution. The thermal functional \(\require{physics} \omega(A) = \Tr[{\rm e}^{-\beta H}A]/Z\) is no longer a state in any physical sense.

The analytic behaviour is clarified by decomposing the time-evolution factor under complex \(z = t + {\rm i}\alpha\): \[{\rm e}^{{\rm i}(E_n - E_m)z} = {\rm e}^{{\rm i}(\alpha_n - \alpha_m)t} \cdot {\rm e}^{-(\alpha_n - \alpha_m)\alpha} \cdot {\rm e}^{-(\gamma_n - \gamma_m)t} \cdot {\rm e}^{{\rm i}(\gamma_n - \gamma_m)\alpha}. \label{eq:complex-factor}\tag{42}\] The third factor, \({\rm e}^{-(\gamma_n - \gamma_m)t}\), grows exponentially as \(t \to \pm\infty\) on the real axis whenever \(\gamma_n \neq \gamma_m\). This exponential growth on the real boundary is the precise mechanism of failure.

One might attempt to restore the formal symmetry of the spectral series by introducing a complex effective inverse temperature \(\beta_{\rm{eff}} \in \mathbb{C}\), chosen so that \(\Im(\beta_{\rm{eff}})\) absorbs the imaginary parts \(\gamma_n\) of the eigenvalues. Concretely, setting \(\beta_{\rm{eff}} = \beta + {\rm i}\delta\) with \(\delta\gamma_n = \text{const}\) for all \(n\) would cancel the oscillatory factor \({\rm e}^{-{\rm i}\beta\gamma_n}\) in Eq. 41 and give real Boltzmann weights.

This approach fails on three counts. First, it requires a uniform imaginary shift \(\delta\gamma_n = \text{const}\) for all \(n\), which generically cannot hold when the imaginary parts \(\gamma_n\) differ across eigenstates — precisely the case that produces exponential growth in Eq. 42 . Second, even when the formal symmetry of the spectral series is restored, the resulting functional violates positivity (the weights \({\rm e}^{-\beta_{\rm{eff}} E_n}\) are not positive reals) and normalisation (the partition function is complex), so no physical state is defined. Third, the boundedness condition is not recovered: the real-axis exponential growth arises from the factor \({\rm e}^{-(\gamma_n - \gamma_m)t}\), which depends on pairwise differences \(\gamma_n - \gamma_m\) and cannot be removed by a global shift of \(\beta\).

No consensus framework for assigning a KMS-type thermal equilibrium structure to systems with complex spectra currently exists. The obstruction is fundamental: complex Boltzmann weights are incompatible with the positivity and boundedness requirements that define a physical thermal state.

Table 3 contrasts the two failure modes analysed in this section.

The table makes clear that the two failure modes, while both fatal to the KMS framework, are mechanistically distinct. Exceptional points corrupt the geometric structure of the eigensystem, and complex spectra corrupt the analytic behaviour of the thermal weights. Future extensions of the non-Hermitian KMS theory — whether through Jordan-adapted modular flows or through a relaxation of the positivity requirement — will need to address these two obstructions separately.

7 Conclusion↩︎

This paper has addressed a single question: to what extent does the standard KMS characterisation of thermal equilibrium extend to non-Hermitian quantum systems? The answer depends sharply on which structural properties of the Hamiltonian are assumed, and the three routes developed here give a precise map of the terrain.

• Route I: A complete KMS theorem for quasi-Hermitian systems. The central result of the paper is the Spectral KMS Theorem (Thm. 9): under Assum. 1, the \(\eta\)-Gibbs state \(\omega_\eta\) satisfies all three analytic conditions of Def. 1. Specifically, the correlation function \(F_{AB}(z) = \omega_\eta(A\,\sigma_z(B))\) is analytic on the thermal strip \(\mathcal{S}_\beta\), bounded and continuous on its closure \(\overline{\mathcal{S}}_\beta\), and satisfies the boundary relation \(\omega_\eta(\sigma_t(A)\,B) = \omega_\eta(B\,\sigma_{t+i\beta}(A))\) for all \(t \in \mathbb{R}\). The supporting results — positivity and faithfulness of \(\omega_\eta\) (Thm. 3), the \(*\)-automorphism property of \(\sigma_t\) in the \(\eta\)-Hilbert space (Thm. 2), and the trace-class properties of \({\rm e}^{-\beta H}\) (Rem. 4) — confirm that \(\omega_\eta\) is a genuine physical state, not merely a formal thermal-looking functional. The proof is self-contained: the Hadamard three-line theorem is applied to finite partial sums of the spectral series to avoid circular reasoning, and trace-class estimates ensure the absolute convergence of all thermal traces.

  The result is non-trivial despite the existence of the intertwining map \(U = \eta^{1/2}\), which relates \(H\) to the self-adjoint Hamiltonian \(h = UHU^{-1}\). The transported state \(\require{physics} \hat{\omega}(X) = \omega_\eta(U^{-1}XU) = \Tr[{\rm e}^{-\beta h}X\eta]/Z_\eta\) differs from the standard Gibbs state \(\omega_h\) of \(h\) whenever \([\eta, h] \neq 0\), and its KMS property does not follow from the Hermitian theory. The proof in Sec. 3.3 provides the independent verification that this requires.

• Route II: A structural characterisation of quasi-Hermiticity. Under the weaker hypothesis of biorthogonal completeness alone, the formal KMS-type identity \(\omega_{\rm{bi}}(\sigma_t(A)\,B) = \omega_{\rm{bi}}(B\,\sigma_{t+i\beta}(A))\) holds (Prop. 11), together with the strip analyticity and time-translation invariance of \(\omega_{\rm{bi}}\). However, positivity \(\omega_{\rm{bi}}(A^\dagger A) \geq 0\) fails generically, and without it \(\omega_{\rm{bi}}\) is not a physical state. The Biorthogonal KMS Structure Theorem (Thm. 13) closes this gap from both directions: positivity of \(\omega_{\rm{bi}}\) holds if and only if \(H\) is quasi-Hermitian. This provides a metric-free characterisation of quasi-Hermiticity — starting from a property of the thermal functional rather than assuming \(\eta\) a priori — and lies strictly outside the Mostafazadeh–Scholtz similarity-transformation framework.

• Route III: Open systems and quantum detailed balance. For open quantum systems governed by the GKSL master equation, the Fagnola–Umanità standard quantum detailed balance condition (Def. 5) provides a sufficient condition for a KMS-type thermal steady state in terms of the full Lindblad data \(\{H_{\rm{sys}}, L_k\}\). This route is logically independent of Routes I and II: the KMS properties of the steady state depend on the full generator \(\mathcal{L}\), not on the effective non-Hermitian Hamiltonian \(H_{\rm{eff}}\) alone, and the two frameworks cannot be directly compared.

Failure analysis. The KMS framework fails at exceptional points through the polynomial growth Eq. 40 , loss of biorthogonal completeness, and consequent violation of the boundedness condition. For complex spectra, the failure mechanism is different: the exponential growth of the factor \({\rm e}^{-(\gamma_n - \gamma_m)t}\) on the real time axis makes the correlation function unbounded on \(\partial\overline{\mathcal{S}}_\beta\), violating the \(L^\infty\) boundary bounds required by the Hadamard three-line theorem. Both cases remain open problems, and the Tab. 4 below records the precise status of each route and failure mode.

Open problems and outlook. Three directions for future work emerge naturally from the analysis.

• The first is the gap between the Spectral KMS Theorem and the full Haag–Hugenholtz–Winnink theorem in the \(C^*\)-algebraic sense. Closing this gap requires equipping the observable algebra with a \(C^*\)-norm, establishing strong continuity of \(\sigma_t\) in the \(\sigma\)-weak operator topology, and constructing the Tomita–Takesaki modular operator \(\Delta\) for the \(\eta\)-Gibbs state. The last point is particularly significant: \(\Delta\) encodes the entire modular structure and its domain theory, and its construction for quasi-Hermitian systems would provide the analogue of the KMS modular theory in the non-Hermitian setting.

• The second direction is the extension of Route I to unbounded Hamiltonians. The present paper assumes \(H \in \mathcal{B}(\mathcal{H})\) to ensure norm-convergence of the operator exponential. For physically relevant Hamiltonians — Schrödinger operators, lattice Hamiltonians with infinite-range interactions — the exponential \({\rm e}^{{\rm i}Ht}\) must be defined via Stone’s theorem, and the KMS proof requires domain-theoretic methods along the lines of Ref. das[4]. The biorthogonal structure of the proof should survive this extension, but the analytic estimates need to be reworked in the graph-norm topology.

• The third direction concerns exceptional points and complex spectra. For exceptional points, the natural question is whether a Jordan-block adaptation of the modular flow can be constructed — replacing \({\rm e}^{{\rm i}E_n t}\) by the polynomial-exponential expression Eq. 40 throughout the spectral theory. For complex spectra, the question is whether a consistent thermal framework can be defined by relaxing the positivity requirement or passing to an indefinite-inner-product Hilbert space. Both directions require new conceptual foundations beyond those developed here.

Acknowledgement↩︎

This work was supported by the Research Fund of Yantai University under Grant No. 2222018.

8 Standard KMS proof for Hermitian systems↩︎

We include a self-contained proof for the reader’s convenience and to establish the spectral identities that are used in Sec. 3.

9 Basic properties of pseudo-Hermitian operators↩︎

This appendix collects the two spectral propositions cited in Sec. 2 (Prop. 1) and used throughout the paper. Both propositions follow directly from the pseudo-Hermitian relation \(H^\dagger = \eta H \eta^{-1}\) and the positive-definiteness of \(\eta\).

10 Equivalence of Routes I and II↩︎

This appendix establishes the precise relationship between the two routes. Under the full Assum. (A1), the biorthogonal state \(\omega_{\rm{bi}}\) of Route II coincides with the \(\eta\)-Gibbs state \(\omega_\eta\) of Route I, so the biorthogonal KMS-type identity (Prop. 11) automatically inherits all the properties established for \(\omega_\eta\) in Sec. 3. In finite dimensions, the converse is also true: by the Biorthogonal KMS Structure Theorem (Thm. 13), positivity of \(\omega_{\rm{bi}}\) implies quasi-Hermiticity, so Assum. 1 is in fact equivalent to positivity of \(\omega_{\rm{bi}}\) in the finite-dimensional setting. The present appendix formalises the forward direction, valid in any dimension, and records the status of the converse in infinite dimensions as an open problem.

References↩︎

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1. stlanchen@126.com↩︎

2. luyao26428@s.ytu.edu.cn↩︎

3. hyang@ucas.ac.cn↩︎