Information geometry extends beyond classical statistical models. Thermodynamic geometry and quantum information geometry keep the same themes: states, distinguishability, monotonicity, and convex duality.
Thermodynamic potentials
Equilibrium thermodynamics has a Hessian geometry built from convex potentials.
Definition. For an exponential-family form of equilibrium states, the log-partition function \(\psi(\theta)\) generates expectation coordinates and a Hessian metric \[ g_{ij}=\partial_i\partial_j\psi. \] This metric is the covariance matrix of the sufficient statistics or observables.
Proposition. Thermodynamic susceptibilities are entries of the Fisher/Hessian metric.
Proof. Differentiating expectation parameters \(\eta_i=\partial_i\psi\) with respect to natural parameters gives \(\partial_j\eta_i=\partial_i\partial_j\psi\). In exponential-family thermodynamics this derivative is the covariance of the corresponding observables, i.e. a susceptibility.
Quantum states
Quantum information geometry replaces probability distributions by density matrices.
Definition. A finite-dimensional quantum state is a positive semidefinite matrix \(\rho\) with \(\operatorname{tr}\rho=1\). A quantum channel is a completely positive trace-preserving linear map \(\Phi\).
Remark. Noncommutativity makes tangent directions and likelihood ratios more subtle than in the classical simplex.
Monotone quantum metrics
Unlike the classical case, monotone quantum metrics are not unique.
Classification. Petz's theorem classifies monotone Riemannian metrics on quantum state spaces by operator-monotone functions. The key mechanism is that monotonicity under every quantum channel forces the metric to be represented by an operator mean acting on left and right multiplication by \(\rho\). The symmetric logarithmic derivative metric and the Bogoliubov-Kubo-Mori metric are important examples.
Classical limit
When all density matrices commute, quantum geometry reduces to classical information geometry.
Proposition. On a commuting family of density matrices, every monotone quantum metric restricts to a constant multiple of the classical Fisher metric.
Proof. Commuting density matrices are simultaneously diagonalizable and therefore equivalent to probability vectors. Quantum channels preserving the commutative algebra include classical Markov kernels. Cencov uniqueness then forces the classical restriction to be Fisher up to scale.
Gibbs families and thermodynamic potentials
Let observables \(F_1,\ldots,F_d\) define the Gibbs family \[ p_\theta(x) =\exp\{\theta^\top F(x)-\psi(\theta)\}h(x), \qquad \psi(\theta)=\log\int e^{\theta^\top F}h\,\mathrm{d}\mu. \] Then \[ \eta_i=\partial_i\psi=E_\theta F_i,\qquad g_{ij}=\partial_{ij}\psi=\operatorname{Cov}_\theta(F_i,F_j). \] In thermodynamics, \(\theta\) are intensive variables, \(\eta\) extensive expectations, and the Legendre transform exchanges thermodynamic potentials. Response coefficients such as heat capacity and susceptibility are metric components.
Example (canonical ensemble). With \(p_\beta(x)=e^{-\beta E(x)}/Z(\beta)\), \[ I(\beta)=\operatorname{Var}_\beta(E), \qquad C_V=\frac{\operatorname{Var}_\beta(E)}{k_BT^2}. \] Large Fisher information corresponds to strong thermodynamic response; its growth near a phase transition signals rapidly changing equilibrium laws.
Quantum states and tangent directions
A finite-dimensional quantum state is a density matrix \(\rho\succeq0\), \(\operatorname{tr}\rho=1\). At a faithful state \(\rho\succ0\), tangent vectors are Hermitian trace-zero matrices. A quantum channel is a completely positive trace-preserving map \(\Phi\), and a quantum monotone metric must contract: \[ g_{\Phi(\rho)}(\Phi_*A,\Phi_*A)\le g_\rho(A,A). \]
Noncommutativity creates many monotone metrics
Petz's classification associates monotone quantum metrics with operator-monotone functions \(f\) satisfying \(f(t)=tf(t^{-1})\). Writing left and right multiplication as \(L_\rho(A)=\rho A\), \(R_\rho(A)=A\rho\), one form is \[ g_\rho^{(f)}(A,B) =\operatorname{tr}\!\left[ A\,\{R_\rho^{1/2}f(L_\rho R_\rho^{-1})R_\rho^{1/2}\}^{-1}(B) \right]. \] When all matrices commute, every such metric reduces, up to scale, to the classical Fisher metric. Their differences are purely noncommutative.
Relative entropy and the BKM metric
Quantum relative entropy is \[ D(\rho\|\sigma) =\operatorname{tr}\{\rho(\log\rho-\log\sigma)\}. \] Its Hessian induces the Bogoliubov--Kubo--Mori metric \[ g_\rho^{\mathrm{BKM}}(A,B) =\int_0^\infty \operatorname{tr}\{A(\rho+tI)^{-1}B(\rho+tI)^{-1}\}\,\mathrm{d}t. \] Equivalently, in exponential coordinates it is the Hessian of the quantum log partition function. The data-processing inequality \(D(\Phi\rho\|\Phi\sigma)\le D(\rho\|\sigma)\) implies monotonicity of its local metric.
Quantum Fisher information in estimation
For a parametric state \(\rho_\theta\), the symmetric logarithmic derivative \(L_i\) solves \[ \partial_i\rho_\theta =\frac12(\rho_\theta L_i+L_i\rho_\theta), \] and the SLD Fisher matrix is \[ J_{ij}^{\mathrm{SLD}} =\frac12\operatorname{tr}\rho_\theta(L_iL_j+L_jL_i). \] Any measurement produces a classical model whose Fisher information is bounded by \(J^{\mathrm{SLD}}\). Unlike the scalar classical case, incompatible optimal measurements for different parameters can prevent simultaneous attainment of the matrix bound.
These extensions preserve the central principle of information geometry: distinguishability determines local metric structure. They also show precisely where classical uniqueness ends and operator geometry begins.