Look: it’s GNS Hilbert spaces everywhere!

Florio M. Ciaglia

Universidad Carlos III de Madrid

2026-01-14

Classical models

Parametric statistical model on the measure space \((X,\mu)\): \[ \mathrm{i}\colon\Theta\subseteq\mathbb{R}^{n} \hookrightarrow \mathcal{P}(\Omega) \] \[ (\theta^1,\dots,\theta^n)\equiv \theta \mapsto \mathrm{i}(\theta)= \mathrm{p}_{\theta}(A)=\int_{A}p(\theta;\omega)\,\mathrm{d}\mu(\omega) \]

Score functions: \[ s_{j}(\theta;\omega):=\frac{\partial}{\theta^j}\log p(\theta;\omega)\,\in\;\mathcal{L}^{2}(\Omega,\mathrm{p}_{\theta}) \]

Fisher-Rao metric:

\[ g^{\mathrm{FR}}_{jk}(\theta)=\int_{\Omega} s_{j}(\theta;\omega)\,s_{k}(\theta;\omega)\, p(\theta;\omega)\,d\mu(\omega) = \mathbb{E}_{p_\theta}[s_{j} \,s_{k}] \]

Amari-Čencov tensor:

\[ T_{jkl}^{\mathrm{AC}}(\theta)=\int_{\Omega} s_{j}(\theta;\omega)\,s_{k}(\theta;\omega)\,s_{l}(\theta;\omega)\, p(\theta;\omega)\,d\mu(\omega)= \mathbb{E}_{p_\theta}[s_{j}\, s_{k}\,s_{l}] \]

Pure Quantum models

Parametric model of normalized vectors on the Hilbert space of the system: \[ \mathrm{i}\colon \Lambda\subseteq\mathbb{R}^{n} \hookrightarrow \mathcal{H} \]

\[ (\lambda^1,\dots,\lambda^n)= \lambda \mapsto \mathrm{i}(\lambda)=\psi(\lambda) \in \mathcal{H},\quad \langle \psi(\lambda)|\psi(\lambda)\rangle = 1 \]

Quantum geometric tensor (QGT): \[ Q_{ij}(\lambda) = \langle \partial_i \psi(\lambda), \partial_j \psi(\lambda)\rangle - \langle \partial_i \psi(\lambda), \psi(\lambda)\rangle \langle \psi(\lambda), \partial_j \psi(\lambda)\rangle \]

• Symmetric part: \[ g_{ij}^{\mathrm{FS}}(\lambda)= \Re Q_{ij}(\lambda) \] is essentially the Fubini-Study metric.

• Anti-symmetric part: \[ \omega_{ij}(\lambda)= \Im Q_{ij}(\lambda) \] is a symplectic form.

Faithful Quantum models

Parametric model of invertible density operators on the Hilbert space of the system: \[ \mathrm{i}\colon \Lambda\subseteq\mathbb{R}^{n} \hookrightarrow \mathcal{B}_{1}(\mathcal{H}) \]

\[ (\lambda^1,\dots,\lambda^n)= \lambda \mapsto \mathrm{i}(\lambda)=\rho(\lambda) \in \mathcal{B}_{1}(\mathcal{H}),\quad \rho(\lambda)>0,\quad \mathrm{Tr}_{\mathcal{H}}(\rho(\lambda))=1 \]

Symmetric Logarithmic Derivatives (SLDs):

\[ \frac{\partial \rho(\lambda)}{\partial \lambda^{j}}=\{\mathbf{L}_{j}(\lambda),\rho(\lambda)\}=\frac{1}{2}\left(\mathbf{L}_{j}(\lambda)\,\rho(\lambda) + \rho(\lambda)\,\mathbf{L}_{j}(\lambda)\right) \]

Bures-Helstrom (Quantum Fisher) metric

\[ g_{jk}^{\mathrm{BH}}(\lambda)=\mathrm{Tr}_{\mathcal{H}}\left(\rho(\lambda)\left\{\mathbf{L}_{j}(\lambda), \mathbf{L}_{k}(\lambda)\right\}\right)=\Re \mathrm{Tr}_{\mathcal{H}}\left(\rho(\lambda)\,\mathbf{L}_{j}(\lambda)\, \mathbf{L}_{k}(\lambda)\right) \]