Thermal Quantum Geometric Tensor

• Thermal Quantum Geometric Tensor is a mixed-state extension of the standard QGT, splitting into a real part (Bures metric/quantum Fisher information) and an imaginary part dependent on the framework.
• It employs various methodologies—Uhlmann purification, SLD-based approaches, and adiabatic response—to reveal finite-temperature geometric phenomena and mixed-state ambiguities.
• The tensor’s metric governs statistical distinguishability and optimal parameter estimation while its curvature component connects to observable effects like Hall noise and energy transport.

Searching arXiv for recent and directly relevant papers on thermal/mixed-state quantum geometric tensor. The thermal quantum geometric tensor is a finite-temperature or mixed-state extension of the quantum geometric tensor, the local tensorial object whose real part defines a metric and whose imaginary part defines a curvature-type antisymmetric form. In current arXiv usage, the expression is not tied to a single universally adopted definition. It refers, depending on context, to the gauge-invariant mixed-state QGT obtained from density matrices and purification, whose real and imaginary parts are the Bures metric and the Uhlmann form (Hou et al., 2023); to the finite-temperature quantum Fisher tensor built from the quantum parts of symmetric logarithmic derivatives and its unequal-time generalization (Ji et al., 18 Jul 2025); to the response tensor governing adiabatic quantum thermal machines (Bhandari et al., 2020); and to energy-weighted transport tensors in the thermal channel (Lhachemi et al., 10 Mar 2026). A separate but closely related line of work identifies finite-temperature Hall noise as an observable manifestation of quantum fluctuations of the QGT (Wei et al., 2023).

1. Pure-state antecedent and the finite-temperature extension problem

For normalized pure states $|\psi(R)\rangle$, the standard quantum geometric tensor is

$$Q^{\mathrm{pure}}_{ij}=\langle \partial_i\psi | (1-|\psi\rangle\langle\psi|) |\partial_j\psi\rangle.$$

Its real part is the Fubini–Study metric, while its imaginary part is $-(i/2)$ times the Berry curvature, with Berry connection $A_i=\langle \psi|\partial_i\psi\rangle$ and curvature $F_{ij}=\partial_i A_j-\partial_j A_i$ (Hou et al., 2023). In Bloch-band language one also writes

$$\mathcal{T}^{(n)}_{ij}(\mathbf{k}) \equiv g^{(n)}_{ij}(\mathbf{k}) + i\, \Omega^{(n)}_{ij}(\mathbf{k}),$$

with

$$g^{(n)}_{ij}(\mathbf{k}) = \mathrm{Re}\,\langle \partial_i u_n | \left(1 - |u_n\rangle\langle u_n|\right) | \partial_j u_n \rangle,\quad \Omega^{(n)}_{ij}(\mathbf{k}) = 2\, \mathrm{Im}\,\langle \partial_i u_n | \partial_j u_n \rangle$$

(Wei et al., 2023).

The finite-temperature extension problem arises because thermal equilibrium is described by a density matrix rather than by a ray in projective Hilbert space. The resulting geometry is therefore formulated on the manifold of density operators, or on auxiliary bundles over that manifold, rather than on $CP^{N-1}$. A central theme across the literature is that the real part of the thermal or mixed-state tensor is closely tied to Bures geometry and quantum Fisher information, whereas the status of the imaginary part depends strongly on the chosen framework. In the Uhlmann purification construction it is the Uhlmann form and vanishes identically for ordinary processes (Hou et al., 2023); in the SLD-based finite-temperature tensor it is the mean Uhlmann curvature (Ji et al., 18 Jul 2025); in adiabatic thermal machines it is a Berry-curvature-like antisymmetric response tensor (Bhandari et al., 2020). This terminological plurality is a recurrent source of confusion.

2. Mixed-state QGT from density matrices and purification

A canonical mixed-state construction starts from full-rank density matrices and their purifications. The base manifold is the manifold of full-rank density matrices, denoted $\mathcal{D}^N_N$, and the total purification space is

$$\mathcal{S}_N=\{W\in \mathrm{End}(\mathcal{H}) \mid \mathrm{Tr}(W^\dagger W)=1,\ \mathrm{rank}\,W=N\},$$

with projection $$Q^{\mathrm{pure}}_{ij}=\langle \partial_i\psi | (1-|\psi\rangle\langle\psi|) |\partial_j\psi\rangle.$$0 and gauge symmetry $$Q^{\mathrm{pure}}_{ij}=\langle \partial_i\psi | (1-|\psi\rangle\langle\psi|) |\partial_j\psi\rangle.$$1 for $$Q^{\mathrm{pure}}_{ij}=\langle \partial_i\psi | (1-|\psi\rangle\langle\psi|) |\partial_j\psi\rangle.$$2, so that $$Q^{\mathrm{pure}}_{ij}=\langle \partial_i\psi | (1-|\psi\rangle\langle\psi|) |\partial_j\psi\rangle.$$3 (Hou et al., 2023). The raw local distance on $$Q^{\mathrm{pure}}_{ij}=\langle \partial_i\psi | (1-|\psi\rangle\langle\psi|) |\partial_j\psi\rangle.$$4 is

$$Q^{\mathrm{pure}}_{ij}=\langle \partial_i\psi | (1-|\psi\rangle\langle\psi|) |\partial_j\psi\rangle.$$5

Its symmetric and antisymmetric parts are not gauge invariant by themselves.

Gauge invariance is restored by introducing the Uhlmann connection

$$Q^{\mathrm{pure}}_{ij}=\langle \partial_i\psi | (1-|\psi\rangle\langle\psi|) |\partial_j\psi\rangle.$$6

in the eigenbasis $$Q^{\mathrm{pure}}_{ij}=\langle \partial_i\psi | (1-|\psi\rangle\langle\psi|) |\partial_j\psi\rangle.$$7, together with the Ehresmann connection

$$Q^{\mathrm{pure}}_{ij}=\langle \partial_i\psi | (1-|\psi\rangle\langle\psi|) |\partial_j\psi\rangle.$$8

for $$Q^{\mathrm{pure}}_{ij}=\langle \partial_i\psi | (1-|\psi\rangle\langle\psi|) |\partial_j\psi\rangle.$$9 (Hou et al., 2023). The gauge-invariant real part is the authors’ “Uhlmann metric”

$-(i/2)$0

which is independent of the fiber $-(i/2)$1 and equals the Bures metric. The corresponding eigenbasis expression is

$-(i/2)$2

Equivalently,

$-(i/2)$3

with $-(i/2)$4 the symmetric-logarithmic-derivative quantum Fisher information matrix defined by $-(i/2)$5 (Hou et al., 2023).

The gauge-invariant imaginary part is the Uhlmann form,

$-(i/2)$6

Its geometric role is not the same as the Berry curvature of pure-state geometry. The Uhlmann form is not proportional to the curvature $-(i/2)$7, and for finite-dimensional systems under ordinary physical processes—described in the paper as trace-preserving, unitary-or-Markovian—it vanishes identically (Hou et al., 2023). This sharply distinguishes mixed-state thermal geometry from the Kähler geometry of pure states. The same paper attributes this difference to the fact that $-(i/2)$8 lacks the Kähler structure of $-(i/2)$9 and that the Uhlmann bundle is topologically trivial.

3. Gibbs states, local geometry, and the zero-temperature limit

For thermal equilibrium states,

$A_i=\langle \psi|\partial_i\psi\rangle$0

and in the eigenbasis $A_i=\langle \psi|\partial_i\psi\rangle$1 of $A_i=\langle \psi|\partial_i\psi\rangle$2 one has

$A_i=\langle \psi|\partial_i\psi\rangle$3

In this setting the Bures metric takes the spectral form

$A_i=\langle \psi|\partial_i\psi\rangle$4

and the $A_i=\langle \psi|\partial_i\psi\rangle$5 representation

$A_i=\langle \psi|\partial_i\psi\rangle$6

(Hou et al., 2023). These formulas separate basis geometry from population changes: off-diagonal terms encode the geometry of the eigenvectors, while diagonal terms encode derivatives of the thermal populations.

For thermal states the Uhlmann connection simplifies to

$A_i=\langle \psi|\partial_i\psi\rangle$7

and the Uhlmann form remains

$A_i=\langle \psi|\partial_i\psi\rangle$8

for ordinary thermal processes (Hou et al., 2023).

For two-level systems, writing

$A_i=\langle \psi|\partial_i\psi\rangle$9

the Bures distance becomes

$F_{ij}=\partial_i A_j-\partial_j A_i$0

hence

$F_{ij}=\partial_i A_j-\partial_j A_i$1

For the spin-$F_{ij}=\partial_i A_j-\partial_j A_i$2 paramagnet with

$F_{ij}=\partial_i A_j-\partial_j A_i$3

the metric components are

$F_{ij}=\partial_i A_j-\partial_j A_i$4

As $F_{ij}=\partial_i A_j-\partial_j A_i$5, these approach the Fubini–Study metric on $F_{ij}=\partial_i A_j-\partial_j A_i$6; as $F_{ij}=\partial_i A_j-\partial_j A_i$7, $F_{ij}=\partial_i A_j-\partial_j A_i$8 as the state approaches the center of the Bloch ball (Hou et al., 2023). In a two-dimensional two-band model,

$F_{ij}=\partial_i A_j-\partial_j A_i$9

the explicit $$\mathcal{T}^{(n)}_{ij}(\mathbf{k}) \equiv g^{(n)}_{ij}(\mathbf{k}) + i\, \Omega^{(n)}_{ij}(\mathbf{k}),$$0, $$\mathcal{T}^{(n)}_{ij}(\mathbf{k}) \equiv g^{(n)}_{ij}(\mathbf{k}) + i\, \Omega^{(n)}_{ij}(\mathbf{k}),$$1, and $$\mathcal{T}^{(n)}_{ij}(\mathbf{k}) \equiv g^{(n)}_{ij}(\mathbf{k}) + i\, \Omega^{(n)}_{ij}(\mathbf{k}),$$2 exhibit saddle structures and nonzero off-diagonal components, thereby contrasting sharply with the paramagnetic example (Hou et al., 2023).

A central structural result is the zero-temperature correspondence. For a finite-dimensional Gibbs state with nondegenerate ground state $$\mathcal{T}^{(n)}_{ij}(\mathbf{k}) \equiv g^{(n)}_{ij}(\mathbf{k}) + i\, \Omega^{(n)}_{ij}(\mathbf{k}),$$3, one has

$$\mathcal{T}^{(n)}_{ij}(\mathbf{k}) \equiv g^{(n)}_{ij}(\mathbf{k}) + i\, \Omega^{(n)}_{ij}(\mathbf{k}),$$4

with $$\mathcal{T}^{(n)}_{ij}(\mathbf{k}) \equiv g^{(n)}_{ij}(\mathbf{k}) + i\, \Omega^{(n)}_{ij}(\mathbf{k}),$$5, hence

$$\mathcal{T}^{(n)}_{ij}(\mathbf{k}) \equiv g^{(n)}_{ij}(\mathbf{k}) + i\, \Omega^{(n)}_{ij}(\mathbf{k}),$$6

The real part of the mixed-state tensor therefore reduces to the Fubini–Study metric of the ground state in the zero-temperature limit (Hou et al., 2023). The same framework also yields a Pythagorean-like relation,

$$\mathcal{T}^{(n)}_{ij}(\mathbf{k}) \equiv g^{(n)}_{ij}(\mathbf{k}) + i\, \Omega^{(n)}_{ij}(\mathbf{k}),$$7

which decomposes the raw distance in the total space into base and fiber contributions (Hou et al., 2023).

4. SLD-based finite-temperature tensors and time dependence

A different but closely allied construction starts from the thermal density matrix

$$\mathcal{T}^{(n)}_{ij}(\mathbf{k}) \equiv g^{(n)}_{ij}(\mathbf{k}) + i\, \Omega^{(n)}_{ij}(\mathbf{k}),$$8

and from the symmetric logarithmic derivative one-form $$\mathcal{T}^{(n)}_{ij}(\mathbf{k}) \equiv g^{(n)}_{ij}(\mathbf{k}) + i\, \Omega^{(n)}_{ij}(\mathbf{k}),$$9 defined by the Lyapunov equation

$$g^{(n)}_{ij}(\mathbf{k}) = \mathrm{Re}\,\langle \partial_i u_n | \left(1 - |u_n\rangle\langle u_n|\right) | \partial_j u_n \rangle,\quad \Omega^{(n)}_{ij}(\mathbf{k}) = 2\, \mathrm{Im}\,\langle \partial_i u_n | \partial_j u_n \rangle$$0

In the eigenbasis of $$g^{(n)}_{ij}(\mathbf{k}) = \mathrm{Re}\,\langle \partial_i u_n | \left(1 - |u_n\rangle\langle u_n|\right) | \partial_j u_n \rangle,\quad \Omega^{(n)}_{ij}(\mathbf{k}) = 2\, \mathrm{Im}\,\langle \partial_i u_n | \partial_j u_n \rangle$$1,

$$g^{(n)}_{ij}(\mathbf{k}) = \mathrm{Re}\,\langle \partial_i u_n | \left(1 - |u_n\rangle\langle u_n|\right) | \partial_j u_n \rangle,\quad \Omega^{(n)}_{ij}(\mathbf{k}) = 2\, \mathrm{Im}\,\langle \partial_i u_n | \partial_j u_n \rangle$$2

where $$g^{(n)}_{ij}(\mathbf{k}) = \mathrm{Re}\,\langle \partial_i u_n | \left(1 - |u_n\rangle\langle u_n|\right) | \partial_j u_n \rangle,\quad \Omega^{(n)}_{ij}(\mathbf{k}) = 2\, \mathrm{Im}\,\langle \partial_i u_n | \partial_j u_n \rangle$$3 (Ji et al., 18 Jul 2025). The diagonal term is the “classical” part and the off-diagonal term is the “quantum” part.

The static finite-temperature quantum geometric tensor is then defined as

$$g^{(n)}_{ij}(\mathbf{k}) = \mathrm{Re}\,\langle \partial_i u_n | \left(1 - |u_n\rangle\langle u_n|\right) | \partial_j u_n \rangle,\quad \Omega^{(n)}_{ij}(\mathbf{k}) = 2\, \mathrm{Im}\,\langle \partial_i u_n | \partial_j u_n \rangle$$4

with

$$g^{(n)}_{ij}(\mathbf{k}) = \mathrm{Re}\,\langle \partial_i u_n | \left(1 - |u_n\rangle\langle u_n|\right) | \partial_j u_n \rangle,\quad \Omega^{(n)}_{ij}(\mathbf{k}) = 2\, \mathrm{Im}\,\langle \partial_i u_n | \partial_j u_n \rangle$$5

Its spectral representation is

$$g^{(n)}_{ij}(\mathbf{k}) = \mathrm{Re}\,\langle \partial_i u_n | \left(1 - |u_n\rangle\langle u_n|\right) | \partial_j u_n \rangle,\quad \Omega^{(n)}_{ij}(\mathbf{k}) = 2\, \mathrm{Im}\,\langle \partial_i u_n | \partial_j u_n \rangle$$6

Here $$g^{(n)}_{ij}(\mathbf{k}) = \mathrm{Re}\,\langle \partial_i u_n | \left(1 - |u_n\rangle\langle u_n|\right) | \partial_j u_n \rangle,\quad \Omega^{(n)}_{ij}(\mathbf{k}) = 2\, \mathrm{Im}\,\langle \partial_i u_n | \partial_j u_n \rangle$$7 is the pullback of the Bures metric, while $$g^{(n)}_{ij}(\mathbf{k}) = \mathrm{Re}\,\langle \partial_i u_n | \left(1 - |u_n\rangle\langle u_n|\right) | \partial_j u_n \rangle,\quad \Omega^{(n)}_{ij}(\mathbf{k}) = 2\, \mathrm{Im}\,\langle \partial_i u_n | \partial_j u_n \rangle$$8 is the mean Uhlmann curvature (Ji et al., 18 Jul 2025).

The same work introduces an unequal-time or time-dependent finite-temperature tensor,

$$g^{(n)}_{ij}(\mathbf{k}) = \mathrm{Re}\,\langle \partial_i u_n | \left(1 - |u_n\rangle\langle u_n|\right) | \partial_j u_n \rangle,\quad \Omega^{(n)}_{ij}(\mathbf{k}) = 2\, \mathrm{Im}\,\langle \partial_i u_n | \partial_j u_n \rangle$$9

with symmetric and antisymmetric parts

$CP^{N-1}$0

At $CP^{N-1}$1 one recovers the static quantities, while in the gapped zero-temperature limit $CP^{N-1}$2 one has

$CP^{N-1}$3

namely the usual pure-state QGT (Ji et al., 18 Jul 2025).

This SLD-based formalism is tied to response theory through a fluctuation–dissipation generating function,

$CP^{N-1}$4

which produces finite-temperature sum rules. In particular,

$CP^{N-1}$5

and

$CP^{N-1}$6

These relations extend mixed-state Fisher and Uhlmann sum rules to finite temperature within a single fluctuation–dissipation framework (Ji et al., 18 Jul 2025).

A significant distinction is that this SLD-based tensor does not coincide at finite temperature with the so-called “Columbia” definition

$CP^{N-1}$7

The two definitions agree at $CP^{N-1}$8 but differ at finite $CP^{N-1}$9, especially on diagonal and gauge issues (Ji et al., 18 Jul 2025). This is one of the clearest finite-temperature ambiguities in the subject.

5. Observable consequences: distinguishability, sum rules, and Hall noise

The real part of the mixed-state tensor has direct operational meaning. Because $\mathcal{D}^N_N$0, the Bures metric governs local statistical distinguishability and optimal parameter-estimation precision in thermal states (Hou et al., 2023). The same work connects it to fidelity susceptibility and Uhlmann fidelity, and notes that in Gibbs ensembles fidelity can be written in terms of partition functions; when the parameter couples as a Zeeman term, the fidelity susceptibility can be inferred from magnetic susceptibility. This provides an experimentally motivated route to the real part of the thermal QGT.

The SLD-based finite-temperature tensor also admits optical and magneto-optical access through sum rules. Besides the conductivity integrals already quoted, the orbital magnetization correlator

$\mathcal{D}^N_N$1

obeys

$\mathcal{D}^N_N$2

leading to the finite-temperature magnetic circular dichroism sum rule

$\mathcal{D}^N_N$3

(Ji et al., 18 Jul 2025). In this sense, finite-temperature geometry is encoded in experimentally accessible dynamical response.

A different observable route emerges from current noise in time-reversal-invariant systems. For Bloch bands, one may define the operator-level fluctuation of a QGT component $\mathcal{D}^N_N$4 by

$\mathcal{D}^N_N$5

In the two-dimensional two-band case, the thermally weighted curvature fluctuation

$\mathcal{D}^N_N$6

is identified as the thermal manifestation of the quantum fluctuation of the QGT (Wei et al., 2023). The thermal Hall noise linear in the applied field is purely intrinsic and controlled by the Berry curvature dipole,

$\mathcal{D}^N_N$7

whereas the second-order Hall noise contains both extrinsic and intrinsic pieces,

$\mathcal{D}^N_N$8

The $\mathcal{D}^N_N$9-independent intrinsic term is the experimentally accessible signature of QGT fluctuation (Wei et al., 2023). This result should not be conflated with the Uhlmann-form statement $$\mathcal{S}_N=\{W\in \mathrm{End}(\mathcal{H}) \mid \mathrm{Tr}(W^\dagger W)=1,\ \mathrm{rank}\,W=N\},$$0: the Hall-noise construction probes thermally weighted fluctuations of Bloch-band geometry rather than the imaginary part of the mixed-state Uhlmann QGT.

6. Related specialized constructions and principal distinctions

In adiabatic quantum thermal machines, the geometric object called the thermal geometric tensor is the response matrix $$\mathcal{S}_N=\{W\in \mathrm{End}(\mathcal{H}) \mid \mathrm{Tr}(W^\dagger W)=1,\ \mathrm{rank}\,W=N\},$$1 defined on an extended parameter manifold containing both slow control parameters and the temperature bias,

$$\mathcal{S}_N=\{W\in \mathrm{End}(\mathcal{H}) \mid \mathrm{Tr}(W^\dagger W)=1,\ \mathrm{rank}\,W=N\},$$2

Its symmetric and antisymmetric parts are

$$\mathcal{S}_N=\{W\in \mathrm{End}(\mathcal{H}) \mid \mathrm{Tr}(W^\dagger W)=1,\ \mathrm{rank}\,W=N\},$$3

with the antisymmetric part

$$\mathcal{S}_N=\{W\in \mathrm{End}(\mathcal{H}) \mid \mathrm{Tr}(W^\dagger W)=1,\ \mathrm{rank}\,W=N\},$$4

identified as a Berry curvature, and the symmetric part acting as a Riemannian metric controlling dissipation (Bhandari et al., 2020). Geometric heat pumping is expressed as a line or surface integral,

$$\mathcal{S}_N=\{W\in \mathrm{End}(\mathcal{H}) \mid \mathrm{Tr}(W^\dagger W)=1,\ \mathrm{rank}\,W=N\},$$5

while the entropy production rate is quadratic in velocities and depends only on the symmetric part. Under time-reversal symmetry the geometric contribution to work satisfies $$\mathcal{S}_N=\{W\in \mathrm{End}(\mathcal{H}) \mid \mathrm{Tr}(W^\dagger W)=1,\ \mathrm{rank}\,W=N\},$$6, and in the adiabatic limit purely geometric machines can approach Carnot efficiency (Bhandari et al., 2020). This is a response-theoretic thermal tensor, not a density-matrix QGT in the Uhlmann sense.

A further specialized usage appears in a zero-temperature transport framework for clean band insulators. There the generalized time-dependent QGT is built from projected particle, energy, and heat polarization operators, and the thermal channel at equal time defines a thermal QGT

$$\mathcal{S}_N=\{W\in \mathrm{End}(\mathcal{H}) \mid \mathrm{Tr}(W^\dagger W)=1,\ \mathrm{rank}\,W=N\},$$7

with

$$\mathcal{S}_N=\{W\in \mathrm{End}(\mathcal{H}) \mid \mathrm{Tr}(W^\dagger W)=1,\ \mathrm{rank}\,W=N\},$$8

and

$$\mathcal{S}_N=\{W\in \mathrm{End}(\mathcal{H}) \mid \mathrm{Tr}(W^\dagger W)=1,\ \mathrm{rank}\,W=N\},$$9

The tensor is positive semidefinite, has a Gram representation, and in two dimensions satisfies the thermal trace condition

$$Q^{\mathrm{pure}}_{ij}=\langle \partial_i\psi | (1-|\psi\rangle\langle\psi|) |\partial_j\psi\rangle.$$00

(Lhachemi et al., 10 Mar 2026). The same paper explicitly states that a finite-temperature geometric generalization is left open, so this thermal-channel tensor is distinct from finite-temperature density-matrix geometry.

A broader geometric thermodynamics program adapts the TQGT language to a contact-manifold and principal-bundle setting. On the Gibbs manifold the real part is taken to be the Bures–Wasserstein metric, which for trace-one density operators coincides with the Uhlmann/Bures metric and the SLD Fisher metric,

$$Q^{\mathrm{pure}}_{ij}=\langle \partial_i\psi | (1-|\psi\rangle\langle\psi|) |\partial_j\psi\rangle.$$01

while the imaginary part is identified with the curvature of a principal connection on thermodynamic labels,

$$Q^{\mathrm{pure}}_{ij}=\langle \partial_i\psi | (1-|\psi\rangle\langle\psi|) |\partial_j\psi\rangle.$$02

(Tejero et al., 13 Nov 2025). In that framework, quasistatic processes minimize geodesic length, cyclic holonomy quantifies irreversibility, and the divergence of Bures–Wasserstein geodesic length toward rank-deficient states is presented as a geometric statement of the third law (Tejero et al., 13 Nov 2025). This suggests an overview between mixed-state information geometry and geometric thermodynamics, but it should still be distinguished from the Uhlmann-form construction of mixed-state QGT.

The modern literature therefore uses “thermal quantum geometric tensor” for several related but non-equivalent objects. The common invariant content is the persistence of metric-curvature decompositions beyond pure states, the central role of Bures or SLD geometry in the real part, and the appearance of experimentally accessible finite-temperature signatures through susceptibility, conductivity, pumping, and noise. The principal conceptual fault lines concern the imaginary part—Uhlmann form, mean Uhlmann curvature, Berry-curvature-like adiabatic response, or heat-magnetization-fixed transport curvature—and the choice of state space on which the tensor is defined.