---
title: Fermi-Dirac Fisher Info in Kinetic & Quantum Models
url: https://www.emergentmind.com/topics/fermi-dirac-type-fisher-information
type: topic
---

# Fermi-Dirac Fisher Info in Kinetic & Quantum Models

Searching arXiv for the cited papers to ground the article in current arXiv records.
Fermi-Dirac-type Fisher information denotes a class of information functionals adapted to fermionic statistics, in which the exclusion factor and the Fermi-Dirac entropy replace the linear mobility and Boltzmann-Shannon entropy underlying the classical Fisher information. In the kinetic setting, the characteristic modification is the replacement of \(f\) by \(f(1-\varepsilon f)\) and of \(\log f\) by \(\log \dfrac{f}{1-\varepsilon f}\), leading to functionals such as
\[
\mathcal{I}_\varepsilon(f)=\int_{\mathbb R^d}\frac{|\nabla f|^2}{f(1-\varepsilon f)}\,dv
\]
and its relative counterpart for Fermi-Dirac-Fokker-Planck dynamics [2508.15054]. In the Landau-Fermi-Dirac setting, the corresponding relative Fisher-information-type quantity is built from the logarithmic Fermi variable \(h=\log \dfrac{g}{1-\varepsilon g}\) and the equilibrium drift, rather than from \(\log g\) alone [2012.12998]. In quantum metrology, an operator-theoretic analogue appears for fermionic Gaussian states, where the relevant quantum Fisher information is determined by covariance matrices whose mode occupations obey Fermi-Dirac-type statistics [1912.12313]. Across these contexts, the common feature is that the information functional is matched to fermionic entropy, Pauli blocking, or Fermi-Dirac occupations rather than to classical Maxwell-Boltzmann structure.

## 1. Conceptual scope and defining structures

The most direct modern formulation appears in the study of Fermi-Dirac-Fokker-Planck dynamics, where the natural mobility is
\[
\mm(f)=f(1-\varepsilon f),
\]
the pointwise constraint is
\[
0\le f\le \varepsilon^{-1},
\]
and the entropy density is
\[
U_\varepsilon(f)=\frac{1}{\varepsilon}\left[\varepsilon f\log (\varepsilon f)+(1-\varepsilon f) \log (1-\varepsilon f)\right].
\]
The associated non-relative Fisher information is defined by
\[
\mathcal{I}_\varepsilon(f):=\int_{\R^d} \mm(f) \left|\nabla\frac{\delta\mathcal{E}_{\varepsilon}}{\delta f}\right|^2 \dif v
=\int_{\R^d} \frac{|\nabla f|^2}{ f(1-\varepsilon f)} \dif v,
\]
while the relative free-energy dissipation is
\[
\JJ(f):=\int_{\R^d} \mm(f)\left|\nabla\;\!\frac{\delta\mathcal{H}_\varepsilon}{\delta f}\right|^2 \dif v
=\int_{\R^d}f(1-\varepsilon f)\left|\frac{\nabla f}{f(1-\varepsilon f)}+v\right|^2 \dif v
\]
[2508.15054]. These formulas define the canonical kinetic meaning of the term.

The structural distinction from the classical Fisher information is explicit. When \(\varepsilon=0\), one has
\[
\mm(f)=f,\qquad U_\varepsilon(f)\to f\log f,\qquad \mathcal{I}_0(f)=\int \frac{|\nabla f|^2}{f}.
\]
The fermionic correction therefore consists of replacing the linear mobility \(f\) by \(f(1-\varepsilon f)\) and the logarithmic variable \(\log f\) by
\[
\log \frac{f}{1-\varepsilon f}
\]
[2508.15054]. This introduces a singular response near saturation \(f\to \varepsilon^{-1}\), reflecting the exclusion principle.

A closely related formulation appears in Landau-Fermi-Dirac theory, where the natural logarithmic variable is
\[
h=\log g-\log(1-\varepsilon g)=\log\frac{g}{1-\varepsilon g},
\qquad
\nabla h=\frac{\nabla g}{g(1-\varepsilon g)}.
\]
In that setting, the weighted relative Fisher information is interpreted as
\[
\mathscr F_\varepsilon^{(\gamma)}(g)= \int_{\mathbb R^3} |\nabla h(v)+2b_\varepsilon v|^2\,g(v)\,\langle v\rangle^{\min(\gamma,0)}\,dv,
\]
which vanishes at the Fermi-Dirac equilibrium \(M_\varepsilon\) [2012.12998]. The same variable also underlies the entropy dissipation estimates in the hard-potential Landau-Fermi-Dirac regime [2401.02877].

## 2. Entropy, mobility, and the exclusion principle

The unifying principle is the exact matching between the entropy Hessian and the fermionic mobility. For the Fermi-Dirac entropy density,
\[
U_\varepsilon''(f)=\mm(f)^{-1},
\]
so that
\[
\nabla \frac{\delta \mathcal{E}_\varepsilon}{\delta f}
=U_\varepsilon''(f)\nabla f
=\frac{\nabla f}{f(1-\varepsilon f)}.
\]
Consequently,
\[
\mm(f)\left|\nabla \frac{\delta \mathcal{E}_\varepsilon}{\delta f}\right|^2
=\frac{|\nabla f|^2}{f(1-\varepsilon f)},
\]
which is precisely the fermionic analogue of \(f |\nabla \log f|^2=|\nabla f|^2/f\) [2508.15054].

This structure is mirrored in multi-species quantum kinetic theory, although not there under the name “Fisher information.” For Fermi particles, the statistical factor is
\[
\tau_i(f)=1-f,
\]
the entropy density is
\[
h_i(f)=f\log f-(1-f)\log(1-f),
\]
and the entropy variable is
\[
\delta H_i(f)=\log\frac{f}{1-f}.
\]
The admissibility range is enforced through
\[
h_i(f)=+\infty \quad \text{if } f\notin [0,1].
\]
In the Landau formulation of that paper, the dissipation variable becomes
\[
\widetilde\nabla^{ij}\delta H(F)
= \Pi_{ij}\left( \frac{\nabla_{v_i} f_i}{m_i f_i(1-f_i)} - \frac{\nabla_{v_j} f_j}{m_j f_j(1-f_j)} \right),
\]
which is one of the closest objects there to a Fermi-Dirac-adapted gradient quantity, though it is treated as entropy dissipation rather than as Fisher information proper [2602.09875].

The same paper is important for delimiting the scope of the term. It includes Fermi-Dirac structure in the collision operators, entropy, and GENERIC formulation, but the actual monotonicity theorem for Fisher information is proved only for the classical homogeneous Boltzmann specialization \(\tau_i\equiv 1\). It therefore does not define or prove decay of a genuinely Fermi-Dirac-adapted Fisher information functional [2602.09875]. This distinction separates entropy-dissipation structures that are “Fisher-information-like” from explicitly named Fermi-Dirac Fisher informations.

## 3. Fermi-Dirac-Fokker-Planck theory

The main kinetic model in which the terminology is introduced explicitly is the space-homogeneous Fermi-Dirac-Fokker-Planck equation
\[
\partial_t f=\Delta f+\nabla\cdot[v f(1-\varepsilon f)].
\]
It is also written in gradient-flow form as
\[
\partial_t f=\nabla\cdot\left[\mm(f)\,\nabla\;\!\frac{\delta\mathcal{H}_\varepsilon}{\delta f}\right],
\]
with free energy
\[
\mathcal{H}_\varepsilon(f):= \mathcal{E}_\varepsilon(f) + \frac{1}{2}\int_{\R^d}|v|^2 f \dif v
\]
and first variation
\[
\frac{\delta\mathcal{H}_\varepsilon}{\delta f} =\log\frac{f}{1-\varepsilon f}+\frac{|v|^2}{2}
\]
[2508.15054].

The entropy dissipation identity is
\[
\frac{\dif}{\dif t}\,\mathcal{H}_\varepsilon(f)
=-\int_{\R^d}\mm(f)\left|\nabla\;\!\frac{\delta\mathcal{H}_\varepsilon}{\delta f}\right|^2 \dif v,
\]
so that
\[
-\frac{d}{dt}\mathcal{H}_\varepsilon(f)=\JJ(f).
\]
In this sense, \(\JJ\) is the relative Fermi-Dirac-type Fisher information: it is exactly the free-energy dissipation rate [2508.15054].

The paper establishes that monotonicity is not unconditional. For any \(\varepsilon>0\), there exist \(\alpha\ge 1\) and \(u\in\R^d\) such that, for
\[
\left.f\right|_{t=0}\!(v)=\frac{1}{\varepsilon+e^{\;\!\alpha\;\!|v-u|^2/2}},
\]
one has
\[
\frac{\dif}{\dif t}\bigg|_{t=0}\JJ(f)>0.
\]
By contrast, if the initial datum satisfies
\[
0\le\left.f\right|_{t=0}\le\mbe,\qquad
\mbe(v):=\frac{1}{\varepsilon+\beta\;\!e^{\;\!|v|^2/2}},
\qquad
0\le4\varepsilon\le\beta,
\]
then
\[
\frac{1}{2}\frac{\dif}{\dif t}\,\JJ(f) \le -\left(1-\frac{4\varepsilon}{\beta}\right)\JJ(f),
\]
so \(\JJ\) is nonincreasing and decays exponentially if \(4\varepsilon<\beta\) [2508.15054].

This conditional monotonicity contrasts with the classical McKean-Toscani mechanism. The obstruction comes from the nonlinear mobility \(\mm(f)=f(1-\varepsilon f)\), which generates mixed-sign terms absent at \(\varepsilon=0\). The paper identifies a sufficient pointwise condition
\[
F(t,v):=1 -2\varepsilon f -\frac{9\varepsilon}{4}|v|^2\;\!\mm\ge0,
\]
under which the decay estimate follows [2508.15054]. A plausible implication is that Fermi-Dirac-type Fisher information is dynamically more sensitive to amplitude constraints than its classical counterpart.

## 4. Landau-Fermi-Dirac and entropy-dissipation estimates

For the spatially homogeneous Landau-Fermi-Dirac equation, the collision operator contains the Pauli factor directly:
\[
Q_{\gamma,\mathrm{LFD}}(f,f)(v) = \nabla_v\cdot \int_{\mathbb{R}^3} |v-w|^{2+\gamma}\,\Pi(v-w)\,
\Bigl( f(w)(1-\varepsilon f(w))\,\nabla f(v) - f(v)(1-\varepsilon f(v))\,\nabla f(w) \Bigr)\,dw.
\]
The associated entropy dissipation is
\[
D_{\gamma,\mathrm{LFD}}(f)
= -\int_{\mathbb{R}^3} Q_{\gamma,\mathrm{LFD}}(f,f)(v)\,
\log\!\left(\frac{f(v)}{1-\varepsilon f(v)}\right)\,dv
\]
and can be rewritten as
\[
D_{\gamma,\mathrm{LFD}}(f)
= \frac12 \iint f(v)(1-\varepsilon f(v))\,f(w)(1-\varepsilon f(w))\,|v-w|^{2+\gamma}\,\Pi(v-w)
\left( \frac{\nabla f}{f(1-\varepsilon f)}(v) - \frac{\nabla f}{f(1-\varepsilon f)}(w) \right)^2 \,dv\,dw \ge 0
\]
[2401.02877].

This formula makes the natural Fermi-Dirac-type Fisher quantity explicit:
\[
I_{\mathrm{FD}}(f)
:= \int_{\mathbb{R}^3}\frac{|\nabla f(v)|^2}{f(v)(1-\varepsilon f(v))}\,dv
= \int_{\mathbb{R}^3} f(v)(1-\varepsilon f(v))
\left| \nabla\log\!\left(\frac{f(v)}{1-\varepsilon f(v)}\right) \right|^2\,dv.
\]
The paper does not isolate this as a named standalone definition, but treats it as the natural object controlled by entropy dissipation [2401.02877].

A more refined relative version is centered around the Fermi-Dirac equilibrium
\[
M_\varepsilon(v)=\frac{a_\varepsilon e^{-b_\varepsilon |v|^2}}{1+\varepsilon a_\varepsilon e^{-b_\varepsilon |v|^2}},
\]
which satisfies
\[
\nabla \log\!\left(\frac{M_\varepsilon}{1-\varepsilon M_\varepsilon}\right)=-2b_\varepsilon v.
\]
Accordingly, the relative Fermi-Dirac Fisher information is
\[
I_{\mathrm{FD,rel}}(f|M_\varepsilon)
= \int_{\mathbb{R}^3} f(1-\varepsilon f)
\left| \nabla\log\!\left(\frac{f}{1-\varepsilon f}\right)+2b_\varepsilon v \right|^2\,dv
\]
[2401.02877].

The hard-potential analysis proves that, under normalization, smoothness, and the positivity gap
\[
1-\varepsilon f(v)\ge k_0>0,
\]
entropy dissipation controls this relative quantity. One form given is
\[
\int_{\mathbb{R}^3} f(v)(1-\varepsilon f(v)) \left| \frac{\nabla f(v)}{f(v)(1-\varepsilon f(v))}+2b_\varepsilon v \right|^2\,dv
- 12\varepsilon^2 \max(\|f\|_\infty,\|M_\varepsilon\|_\infty)^2 \|f-M_\varepsilon\|_{L^1(\mathbb{R}^3)}^2
\le B_f D_{\gamma,\mathrm{LFD}}(f)
\]
[2401.02877]. The same paper emphasizes that the proof avoids direct use of entropy in the lower bounds, unlike earlier methods.

An earlier Landau-Fermi-Dirac paper uses a slightly different but closely related notation. There the weighted relative Fisher information is
\[
\mathscr F_\varepsilon^{(\gamma)}(g)= \int_{\mathbb R^3} |\nabla h(v)+2b_\varepsilon v|^2\,g(v)\,\langle v\rangle^{\min(\gamma,0)}\,dv,
\qquad
h=\log\frac{g}{1-\varepsilon g},
\]
and Proposition 2.12 shows that \(D_\varepsilon(g)\) controls a quantity that “has to be interpreted as a weighted relative Fisher information” for \(h\) [2012.12998]. That paper further derives the comparison
\[
\mathscr F_\varepsilon(g)\ge 2b_\varepsilon H_\varepsilon(g\,|\,M_\varepsilon),
\]
which feeds into exponential convergence to equilibrium for hard potentials [2012.12998].

## 5. Statistical and geometric interpretations

In the information-geometric treatment of ideal quantum gases, Fermi-Dirac-type Fisher information is the Fisher-Rao metric of the grand-canonical fermionic exponential family. The microstate is an occupation-number configuration \(x=\{x_i\}\) with
\[
x_i\in\{0,1\},
\]
the sufficient statistics are total energy and total particle number,
\[
a_1(x)=\sum_i \epsilon_i x_i,\qquad a_2(x)=\sum_i x_i,
\]
and the natural parameters are
\[
\lambda^1=\beta,\qquad \lambda^2=-\beta\mu.
\]
The grand partition function is
\[
Z_{FD}(\lambda)=\prod_i \left(1+e^{-\lambda^1\epsilon_i}e^{-\lambda^2}\right),
\]
and the Fisher-Rao metric is
\[
g_{\mu\nu} = \int dx\, \rho(x|\lambda)
\frac{\partial\log \rho(x|\lambda)}{\partial\lambda^\mu}
\frac{\partial\log \rho(x|\lambda)}{\partial\lambda^\nu}.
\]
Equivalently,
\[
g_{\mu\nu}=C_{\mu\nu}
\]
is the covariance matrix of \((E,N)\), and
\[
g_{\mu\nu} = -\frac{\partial^2 F}{\partial\lambda^\mu\partial\lambda^\nu}
= \frac{\partial^2\log Z}{\partial\lambda^\mu\partial\lambda^\nu}
\]
[2103.00935].

In the continuum density-of-states approximation \(\mathsf G(\epsilon)=\kappa\epsilon^\eta\), the fermionic metric components are given explicitly by
\[
g_{11} = -\kappa \frac{\Gamma(\eta+3)}{\beta^{\eta+3}}Li(-\xi,\eta+2),\qquad
g_{12}= -\kappa \frac{\Gamma(\eta+2)}{\beta^{\eta+2}}Li(-\xi,\eta+1),\qquad
g_{22} = -\kappa \frac{\Gamma(\eta+1)}{\beta^{\eta+1}}Li(-\xi,\eta)
\]
in the paper’s notation [2103.00935]. There, “Fermi-Dirac-type Fisher information” means the parametric Fisher information of the grand-canonical family, not a gradient functional in physical space.

A different geometric perspective arises for finite-dimensional mixed quantum states. The co-adjoint-orbit formalism defines the symmetric logarithmic derivative by
\[
dp=\{\mathrm d_\ell p,p\},
\]
and the Fisher tensor by
\[
\mathfrak F_p=\mathrm{Tr}\big[p\,\mathrm d_\ell p\otimes \mathrm d_\ell p\big].
\]
Although that framework is not specialized to fermions, it is directly applicable to mixed states whose spectra are Fermi-Dirac-type occupation probabilities. The paper shows that the symmetric part of \(\mathfrak F\) is the quantum Fisher metric and the antisymmetric part is a pullback of the Kostant-Kirillov-Souriau symplectic form [1501.00054]. This suggests a broader usage of the term in which Fermi-Dirac-type Fisher information refers to the quantum Fisher geometry of density operators parametrized by Fermi-Dirac occupations.

## 6. Fermionic Gaussian states and quantum Fisher information

For fermionic Gaussian states, the operator-theoretic quantum Fisher information admits a closed form in terms of the covariance matrix. A fermionic Gaussian state is
\[
\rho=\frac{e^{-\frac{i}{4}\,\boldsymbol{\omega}^T \Omega \boldsymbol{\omega}}}{Z},
\]
with covariance matrix
\[
\Gamma_{jk}=\frac12 \operatorname{Tr}\!\left(\rho[\omega_j,\omega_k]\right),
\qquad
\Gamma=\tanh\!\left(i\frac{\Omega}{2}\right).
\]
If \(\Omega\) is brought to canonical block form, the eigenvalues of \(\Gamma\) are
\[
\gamma_k=\tanh\!\left(\frac{\Omega_k}{2}\right),\qquad |\gamma_k|\le 1.
\]
In thermal or Gibbs settings, these \(\gamma_k\) are directly related to Fermi-Dirac occupations [1912.12313].

The symmetric logarithmic derivative for a parameter \(\lambda\) is quadratic in Majorana operators,
\[
L=\frac12\,\boldsymbol{\omega}^T K\,\boldsymbol{\omega} +\frac12\operatorname{Tr}(K\Gamma),
\]
where \(K\) satisfies the Lyapunov-type equation
\[
\dot\Gamma=\Gamma K\Gamma-K.
\]
The resulting quantum Fisher information is
\[
J=-\frac12\,\operatorname{Tr}(K\dot\Gamma)
= -\frac12\,\operatorname{Tr}\!\left[ \dot\Gamma(\operatorname{Ad}_\Gamma-\mathbb 1)^{-1}(\dot\Gamma) \right],
\]
and in the eigenbasis of \(\Gamma\),
\[
J= \frac12\sum_{j,k}\frac{|(\dot\Gamma)_{jk}|^2}{1-\gamma_j\gamma_k}.
\]
For multiple parameters,
\[
J_{\mu\nu}
= -\frac12\,\operatorname{Tr}\!\left[ \partial_\mu\Gamma\, (\operatorname{Ad}_\Gamma-\mathbb 1)^{-1} (\partial_\nu\Gamma) \right]
\]
[1912.12313].

The thermal specialization makes the connection to Fermi-Dirac statistics explicit. For a quadratic fermionic Hamiltonian
\[
H=\frac{i}{4}\,\boldsymbol{\omega}^T G\,\boldsymbol{\omega},
\qquad
\rho_\beta=\frac{e^{-\beta H}}{Z},
\qquad
\Omega=\beta G,
\]
one has
\[
\gamma_k=\tanh\!\left(\frac{\beta\varepsilon_k}{2}\right)
=1-2n_k,
\qquad
n_k=\frac{1}{e^{\beta\varepsilon_k}+1}.
\]
Thus the sensitivity weights can be rewritten in terms of the fluctuation factor
\[
1-\gamma_k^2=4n_k(1-n_k),
\]
and for diagonal perturbations,
\[
J=\frac18\sum_k \operatorname{sech}^2\!\left(\frac{\beta\varepsilon_k}{2}\right)
\big[\partial_\lambda(\beta\varepsilon_k)\big]^2
\]
[1912.12313]. This is the precise sense in which the paper provides a quantum theory of Fisher information for Fermi-Dirac-type fermionic thermal states.

## 7. Relations, limitations, and common misconceptions

One recurring misconception is that any Fisher information appearing in a Fermi-Dirac kinetic paper is automatically a Fermi-Dirac-type Fisher information. This is not always so. In the multi-species GENERIC paper, the only explicitly defined Fisher information is the classical quantity
\[
\mathcal I(F)=\sum_{i=1}^N I(f_i),
\qquad
I(f)=\int_{\mathbb R^d}\frac{|\nabla f|^2}{f}\,dv,
\]
and the monotonicity theorem is proved only after specializing to \(\tau_i(f_i)=1\), כלומר to the classical Maxwell-Boltzmann case [2602.09875]. The Fermi-Dirac contribution there lies in the entropy variables and dissipation structure, not in a named Fisher functional.

A second misconception is that the fermionic modification always improves monotonicity. The Fermi-Dirac-Fokker-Planck study shows the opposite: monotonicity of \(\JJ\) can fail without an upper bound relative to equilibrium [2508.15054]. By contrast, for the heat equation on a Riemannian manifold the paper proves
\[
\frac{1}{2}\frac{\dif}{\dif t}\,\mathcal{I}_{\varepsilon,M}(f)
= -\int_M \mm(f) \big[|D^2\pe|_{\rm HS}^2 +\mathrm{Ric}\;\!(\nabla\pe,\nabla\pe)\big]
+ \varepsilon\;\!|\nabla f|_M^2|\nabla\pe|_M^2 \dif V_M,
\]
and if \(\mathrm{Ric}\ge c_M\), then
\[
\frac{1}{2}\frac{\dif}{\dif t}\,\mathcal{I}_{\varepsilon,M}(f) \le - c_M\,\mathcal{I}_{\varepsilon,M}(f),
\]
so monotonicity is restored under geometric curvature assumptions [2508.15054]. This suggests that the correct behavior depends strongly on the flow.

A third source of ambiguity is the coexistence of classical, quantum-statistical, and quantum-metrological usages. The term may refer to a gradient functional such as
\[
\int \frac{|\nabla f|^2}{f(1-\varepsilon f)},
\]
a relative kinetic dissipation such as \(\JJ\), a weighted Landau-Fermi-Dirac relative Fisher information, a Fisher-Rao metric on grand-canonical Fermi-Dirac models, or a quantum Fisher information of fermionic Gaussian states [2508.15054], [2012.12998], [2103.00935], [1912.12313]. These are not equivalent objects, though they share the same Fermi-Dirac structural ingredients.

A plausible synthesis is that “Fermi-Dirac-type Fisher information” is best viewed as a family resemblance term: the relevant information quantity is the one canonically paired with fermionic entropy, exclusion-modified mobility, or Fermi-Dirac occupation parameters in the model under study.

## 8. Open directions and broader significance

Several papers point toward a broader theory that is still incomplete. The Fermi-Dirac-Fokker-Planck work explicitly states that the corresponding problem for the full Landau-Fermi-Dirac equation will be treated in future work [2508.15054]. The multi-species GENERIC paper provides Fermi-Dirac entropy variables and dissipation structures, but no genuinely Fermi-Dirac-adapted Fisher information functional or monotonicity theorem [2602.09875]. This suggests an open direction connecting GENERIC dissipation operators with fermionic Fisher functionals built from \(\log \dfrac{f}{1-f}\) or \(f(1-f)\).

In the geometric direction, the ideal-gas information-geometry paper shows that the scalar curvature of the Fermi-Dirac statistical manifold is smooth and negative, even for a non-interacting gas, and remains nonzero in the low-fugacity regime [2103.00935]. This indicates that Fisher geometry detects exchange statistics beyond what classical thermodynamic approximations capture. A plausible implication is that geometric and kinetic notions of Fermi-Dirac-type Fisher information may eventually be related through common fluctuation structures.

In many-body quantum metrology, the fermionic Gaussian-state formulas provide an exact covariance-matrix route to quantum Fisher information for thermal states and non-equilibrium steady states whose occupations are Fermi-Dirac distributed [1912.12313]. This makes the subject relevant not only to kinetic entropy methods but also to quantum sensing and criticality in fermionic many-body systems.

Taken together, the literature supports a precise but context-dependent understanding. In kinetic equations, Fermi-Dirac-type Fisher information is the entropy-dissipation-matched quadratic form generated by the mobility \(f(1-\varepsilon f)\) and the logarithmic Fermi variable \(\log \dfrac{f}{1-\varepsilon f}\) [2508.15054], [2012.12998], [2401.02877]. In information geometry, it is the Fisher-Rao metric of Fermi-Dirac grand-canonical families [2103.00935]. In fermionic quantum estimation, it is the quantum Fisher information of Gaussian or thermal states whose covariance spectra encode Fermi-Dirac occupations [1912.12313]. The term therefore denotes not a single universal formula, but a coherent class of Fisher-information constructions adapted to fermionic statistics.

Source: https://www.emergentmind.com/topics/fermi-dirac-type-fisher-information