Diagnosing Thermality from Geometric Observables

1. Introduction

A mixed quantum state may arise from fundamentally different physical mechanisms. On the one hand, it may describe a system in thermal equilibrium at finite temperature,

$$\rho ={\displaystyle \frac{e^{-\beta H}}{Z}},$$

(1)

where H is the Hamiltonian, $\beta$ the inverse temperature, and $Z=\mathrm{Tr}{e}^{-\beta H}$ the partition function. On the other hand, the same density matrix may result from classical uncertainty at zero temperature, represented as a convex mixture of pure states,

$$\rho =\sum _i{p}_i|{\psi }_i\rangle \langle {\psi }_i|.$$

(2)

Although these two scenarios correspond to radically different physical situations, they are not distinguishable from the density matrix alone, since the decomposition of $\rho$ into pure states is not unique. This raises a fundamental question: can one determine whether a given mixed state is thermal using observable quantities only?

This question is of both conceptual and practical importance. In quantum thermodynamics, cold-atom experiments, and quantum simulation platforms, one often has access only to partial information about a system, typically in the form of expectation values and fluctuations of a limited set of observables. In such situations, identifying whether a state is in thermal equilibrium is essential for assigning a temperature, validating effective descriptions, and applying thermodynamic reasoning [1,2,3,4,5].

More broadly, distinguishing thermal from non-thermal states is central to current research on equilibration and thermalization in isolated quantum systems. In particular, the emergence of thermal behavior from unitary dynamics, as well as its breakdown in integrable or many-body localized systems, hinges on identifying whether a given reduced state is genuinely thermal or merely reproduces some of its spectral features.

Standard approaches to this problem typically rely on fitting the state to a Gibbs ensemble or reconstructing the underlying Hamiltonian. Such procedures may be ambiguous or experimentally demanding, and in general require detailed microscopic information. It is therefore desirable to identify criteria that depend only on directly measurable quantities and do not assume prior knowledge of the Hamiltonian [1,2,3,4,5].

In parallel, quantum information geometry provides a natural framework for characterizing quantum states through their response to parameter variations. The quantum Fisher information (QFI), in particular, quantifies the sensitivity of a state to changes in a parameter and plays a central role in quantum estimation theory [1,2]. For thermal states, the QFI with respect to inverse temperature is known to coincide with the energy variance and to satisfy fluctuation–response relations that are fundamental in equilibrium statistical mechanics.

These observations suggest that geometric quantities may encode information not only about the state itself, but also about its physical origin. In this work, we show that this idea leads to a simple and powerful diagnostic of thermality. Specifically, we demonstrate that thermal states are uniquely characterized by the consistency between geometric observables, such as the quantum Fisher information, and thermodynamic fluctuation relations [1,2,3,4,5].

More precisely, we prove that a family of states is of Gibbs form if and only if the quantum Fisher information coincides with the variance of a $\beta$-independent observable and satisfies the corresponding fluctuation–response relation for all values of the parameter. Generic mixed states, lacking an underlying exponential structure, do not satisfy these conditions.

This result provides a direct and operational criterion to distinguish thermal equilibrium from arbitrary statistical mixtures, based solely on observable quantities. Furthermore, we show that the generator of thermal evolution can be reconstructed from geometric data, yielding a Hamiltonian-free formulation in which the effective Hamiltonian emerges from the structure of the state manifold.

It is worth emphasizing that the present approach is based on the information geometry of thermal states, and is therefore distinct from ground-state geometric approaches such as fidelity susceptibility and the quantum geometric tensor, which probe different aspects of quantum criticality [1,2,3,4,5].

Our findings establish a direct connection between quantum information geometry and equilibrium statistical mechanics, and suggest that geometric observables can serve as probes of the physical origin of quantum states. This perspective opens new avenues for the study of inverse problems in quantum systems, where one seeks to infer not only the state itself but also the mechanisms responsible for its preparation.

2. Geometric Structure and Fluctuations

Let $\rho \left(\theta \right)$ be a smooth family of density operators depending on a real parameter $\theta$. The quantum Fisher information (QFI) associated with $\theta$ is defined as [1,2]

$$F_{\theta }=\mathrm{Tr}\left(\rho L_{\theta }^2\right),$$

(3)

where $L_{\theta }$, a Hermitian operator crucial in quantum metrology , is the symmetric logarithmic derivative (SLD), implicitly defined by the Lyapunov equation

$${\partial }_{\theta }\rho ={\displaystyle \frac{1}{2}}(L_{\theta }\rho +\rho L_{\theta }).$$

(4)

For finite-dimensional systems, the QFI admits the spectral representation

$$F_{\theta }=2\sum _{m,n}{\displaystyle \frac{|\langle m|{\partial }_{\theta }{\rho |n\rangle |}^2}{{\lambda }_m+{\lambda }_n}},$$

(5)

where $\rho ={\sum }_m{\lambda }_m|m\rangle \langle m|$ is the spectral decomposition, and the sum runs over all pairs such that ${\lambda }_m+{\lambda }_n>0$.

Thermal states

We now specialize to the case in which the parameter $\theta$ is the inverse temperature $\beta$, and the state has the Gibbs form

$$\rho \left(\beta \right)={\displaystyle \frac{e^{-\beta H}}{Z}},Z=\mathrm{Tr}{e}^{-\beta H}.$$

(6)

Differentiating with respect to $\beta$ yields

$${\partial }_{\beta }\rho =-H\rho +\rho \langle H\rangle ,$$

(7)

where $\langle H\rangle =\mathrm{Tr}\left(\rho H\right)$.

Since $\rho$ commutes with H, the SLD simplifies to

$$L_{\beta }=-(H-\langle H\rangle ),$$

(8)

as can be verified by direct substitution into the defining equation.

The QFI then takes the form

$$F_{\beta }=\mathrm{Tr}\left[\rho {(H-\langle H\rangle )}^2\right]=\langle H^2\rangle -{\langle H\rangle }^2,$$

(9)

which coincides with the energy variance.

Moreover, differentiating the expectation value of the Hamiltonian gives

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial \beta }}\langle H\rangle & =\mathrm{Tr}\left[\left({\partial }_{\beta }\rho \right)H\right]\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & =-\mathrm{Tr}\left[\rho {(H-\langle H\rangle )}^2\right],\hfill \end{array}$$

(11)

so that the fluctuation–response relation

$$F_{\beta }=-{\displaystyle \frac{\partial }{\partial \beta }}\langle H\rangle$$

(12)

is satisfied.

These identities follow directly from the exponential structure of the Gibbs state and constitute a hallmark of equilibrium statistical mechanics. In particular, they show that, for thermal states, geometric information encoded in the QFI is directly linked to physical fluctuations.

Non-thermal states

For a generic mixed state, the situation is fundamentally different. In general, $\rho$ does not commute with the operator generating its parameter dependence, and the SLD cannot be expressed as a simple observable. As a result, the QFI does not reduce to a variance, and the fluctuation–response relation is not satisfied.

This distinction provides the basis for a geometric diagnostic of thermality: thermal states are precisely those for which geometric and fluctuation quantities are consistently related through the above identities.

Geometric characterization of thermality

We now formalize this observation in a precise statement.

Theorem. Let ${\left\{\rho \left(\beta \right)\right\}}_{\beta \in I}$ be a smooth one-parameter family of full-rank density operators on a finite-dimensional Hilbert space. Let $L_{\beta }$ denote the symmetric logarithmic derivative, and $F_{\beta }$ the associated quantum Fisher information.

Then the following statements are equivalent:

1.

There exists a $\beta$-independent Hermitian operator H such that

$$\rho \left(\beta \right)={\displaystyle \frac{e^{-\beta H}}{\mathrm{Tr}{e}^{-\beta H}}}.$$

(13)

2.

The fluctuation–response relation

$$F_{\beta }=-{\displaystyle \frac{\partial }{\partial \beta }}\langle H\rangle$$

(14)

holds for all $\beta \in I$, with $\langle H\rangle =\mathrm{Tr}\left[\rho \right(\beta \left)H\right]$, and the QFI coincides with the variance of H.

Proof. If $\rho \left(\beta \right)$ has Gibbs form, the above relations follow directly from the explicit expressions derived above.

Conversely, assume that the fluctuation–response relation holds for all $\beta$ and that the QFI coincides with the variance of a Hermitian operator H. Then, by comparison with the SLD definition, it follows that on the support of $\rho$

$$L_{\beta }=-(H-\langle H\rangle ).$$

(15)

Substituting into the defining equation for $L_{\beta }$ yields

$${\partial }_{\beta }\rho =-{\displaystyle \frac{1}{2}}(H\rho +\rho H)+\langle H\rangle \rho .$$

(16)

This operator differential equation has the unique solution

$$\rho \left(\beta \right)={\displaystyle \frac{e^{-\beta H}}{\mathrm{Tr}{e}^{-\beta H}}},$$

(17)

up to an additive constant in H. □

This result shows that the fluctuation–response relation is not merely a property of thermal states, but a complete geometric characterization of their structure.

3. Hamiltonian-Free Geometric Diagnosis of Thermality

The previous formulation assumes knowledge of the Hamiltonian H. We now show that this assumption can be removed, and that thermality can be identified purely from geometric data.

Given a smooth family $\rho \left(\beta \right)$, the symmetric logarithmic derivative $L_{\beta }$ provides a natural, observable-dependent generator of parameter translations. Motivated by the thermal case, we define the effective operator

$$H_{\mathrm{eff}}\left(\beta \right):=-L_{\beta }+{\langle L_{\beta }\rangle }_{\beta }.$$

(18)

This operator plays the role of an emergent Hamiltonian associated with the parameter $\beta$.

Proposition (Hamiltonian-free characterization). If the fluctuation–response relation

$$F_{\beta }=-{\displaystyle \frac{\partial }{\partial \beta }}{\langle H_{\mathrm{eff}}\rangle }_{\beta }$$

(19)

holds for all $\beta$, and $H_{\mathrm{eff}}$ is independent of $\beta$, then $\rho \left(\beta \right)$ is a Gibbs state of the form

$$\rho \left(\beta \right)={\displaystyle \frac{e^{-\beta H_{\mathrm{eff}}}}{\mathrm{Tr}{e}^{-\beta H_{\mathrm{eff}}}}}.$$

(20)

This construction shows that the generator of thermal evolution can be recovered directly from geometric observables, without prior knowledge of the microscopic Hamiltonian. In this sense, the quantum Fisher information and the SLD encode not only fluctuation properties but also the effective dynamical generator underlying the statistical state.

Thermality thus emerges as a geometric consistency condition: a state is thermal if and only if its parameter dependence is generated by a $\beta$-independent operator extracted from the geometry of the state manifold. This result establishes that the Hamiltonian is not an external input but an emergent quantity encoded in the information-geometric structure of the state family.

4. Implications

The above criterion provides an operational distinction between thermal and non-thermal mixed states. A generic statistical mixture does not satisfy the fluctuation–response relation, as it lacks an underlying exponential structure.

Importantly, the criterion relies only on observable quantities:

• Expectation values of H,
• Energy fluctuations,
• Parameter sensitivity encoded in the QFI.

No knowledge of the microscopic preparation of the state is required.

5. Two-Level Systeme

Consider a two-level system with energies $\pm ϵ$. For a thermal state,

$$\rho ={\displaystyle \frac{1}{Z}}\left(\begin{array}{cc}{e}^{-\beta ϵ}& 0\\ 0& e^{\beta ϵ}\end{array}\right).$$

(21)

One finds

$$F_{\beta }={ϵ}^2{sech}^2\left(\beta ϵ\right),$$

(22)

and

$$-{\displaystyle \frac{\partial \langle H\rangle }{\partial \beta }}={ϵ}^2{sech}^2\left(\beta ϵ\right),$$

(23)

confirming the criterion.

In contrast, a generic diagonal mixed state with arbitrary probabilities does not satisfy this relation.

5.1. Simple Diagnostic

We illustrate the thermality criterion with a minimal two-level system.

5.2. Thermal State

Consider a Hamiltonian

$$H=ϵ{\sigma }_z,$$

(24)

with eigenvalues $E_{\pm }=\pm ϵ$. The corresponding thermal state is

$${\rho }_{\mathrm{th}}={\displaystyle \frac{1}{Z}}\left(\begin{array}{cc}{e}^{-\beta ϵ}& 0\\ 0& e^{\beta ϵ}\end{array}\right),$$

(25)

where $Z=2cosh\left(\beta ϵ\right)$.

The expectation value of the energy is

$$\langle H\rangle =-ϵtanh\left(\beta ϵ\right),$$

(26)

and the energy variance is

$$\langle H^2\rangle -{\langle H\rangle }^2={ϵ}^2{sech}^2\left(\beta ϵ\right).$$

(27)

The quantum Fisher information with respect to $\beta$ is

$$F_{\beta }={ϵ}^2{sech}^2\left(\beta ϵ\right),$$

(28)

so that the fluctuation–response relation

$$F_{\beta }=-{\displaystyle \frac{\partial \langle H\rangle }{\partial \beta }}$$

(29)

is exactly satisfied.

5.3. Non-Thermal Mixed State

Consider now a generic diagonal mixed state with the same eigenbasis,

$${\rho }_{\mathrm{mix}}=\left(\begin{array}{cc}p& 0\\ 0& 1-p\end{array}\right),$$

(30)

with $0<p<1$ arbitrary.

Defining $\langle H\rangle =ϵ(p-(1-p\left)\right)$, one finds

$$\langle H^2\rangle -{\langle H\rangle }^2=4{ϵ}^{2}p(1-p).$$

(31)

However, since p is not constrained by an exponential law, there is in general no parameter $\beta$ such that

$$F_{\beta }=-{\displaystyle \frac{\partial \langle H\rangle }{\partial \beta }}$$

(32)

holds consistently.

This example shows that thermality is encoded not in the spectrum alone, but in the consistency relations between fluctuations and parameter sensitivity

5.4. Conclusions

This simple example illustrates the thermality criterion. While both states are mixed and diagonal in the same basis, only the thermal state satisfies the geometric fluctuation–response relation. The quantum Fisher information thus provides a direct diagnostic distinguishing thermal equilibrium from generic statistical mixtures.

6. Discussion and Conclusions

We have shown that geometric observables provide a direct and operational diagnostic of thermality. While the density matrix alone does not reveal whether a mixed state originates from thermal equilibrium or from classical statistical mixing, the quantum Fisher information exposes the underlying structure through fluctuation–response relations. In particular, thermal states are uniquely characterized by the consistency between geometric quantities and thermodynamic fluctuations, a property that generic mixed states do not possess.

This result has both conceptual and practical implications. From a practical perspective, it offers a simple criterion to identify equilibrium states using experimentally accessible quantities, without requiring prior knowledge of the Hamiltonian or a full tomographic reconstruction. This is especially relevant in current experimental settings, such as cold atoms and quantum simulators, where only partial information about the system is typically available.

At a conceptual level, our findings suggest that the quantum Fisher information carries a deeper physical meaning than is usually ascribed to it. Traditionally, the QFI is viewed as a statistical or information-theoretic quantity that quantifies parameter sensitivity and sets bounds on estimation precision. Here, however, it plays a more fundamental role: it acts as a diagnostic of the physical origin of a quantum state. In particular, the equality between the QFI and energy fluctuations, together with the associated fluctuation–response relation, provides a signature of thermal equilibrium.

In this sense, the quantum Fisher information acquires a form of operational “reality”: it is not merely a mathematical construct associated with distinguishability, but an observable quantity that encodes whether a state is consistent with an underlying equilibrium structure. More precisely, it captures the compatibility between the state’s geometric response and its fluctuation properties, thereby revealing whether an exponential (Gibbs) description is possible.

This perspective suggests a broader role for geometric quantities in quantum physics. Rather than serving only as tools for estimation or state discrimination, they may provide direct access to structural properties of physical systems, including their thermodynamic nature. This opens new directions for the study of inverse problems, where one seeks to infer not only the state of a system but also the mechanisms responsible for its preparation.

Future work may explore extensions of the present framework to open quantum systems, non-equilibrium steady states, and many-body systems with emergent thermal behavior. It would also be of interest to investigate whether other geometric quantities exhibit similar diagnostic power, and to what extent these ideas can be implemented in realistic experimental platforms.