Noncommutative Information Geometry and Exponential Families

A quantum exponential family is a smooth family of faithful states whose logarithms are affine in chosen noncommuting observables. Its normalization potential generates expectation coordinates and the Bogoliubov–Kubo–Mori (BKM) metric, while Umegaki relative entropy supplies the associated Bregman divergence. These finite-dimensional statements extend to QFT only after support, operator-domain, and state-normality conditions replace the formal trace exponential.

Required background. Bures, Kubo–Mori, and monotone metrics supply the operator means, contraction properties, and boundary behavior needed to distinguish BKM geometry from other quantum metrics.

Helpful background. Relative modular operators and Connes cocycles supply trace-free comparisons of faithful normal states on a von Neumann algebra.

The chapter’s differentiable-family starting point fixes the normalized tangent shared by the local geometries. The comparison of information geometries then records which metric answers which local question, while the independent validity gates separate positivity, common domains, and regulator matching.

Faithful quantum exponential families

Let $\rho_0>0$ on a finite-dimensional Hilbert space and let $F_a=F_a^\dagger$. Define

$$\rho_\theta :=\exp\!\left( \log\rho_0+\theta^aF_a-\psi(\theta) \right), \qquad \psi(\theta) :=\log\operatorname{Tr} e^{\log\rho_0+\theta^aF_a}.$$

Every finite $\theta$ gives a strictly positive density matrix. Differentiating an operator exponential requires the Duhamel formula, not the commuting rule. With $\widehat F_a=F_a-\eta_a\mathbf1$,

$$\partial_a\rho_\theta =\int_0^1\rho_\theta^u\widehat F_a \rho_\theta^{1-u}\,du, \qquad \eta_a:=\partial_a\psi =\operatorname{Tr}(\rho_\theta F_a).$$

The Hessian is

$$g_{ab}^{\rm BKM} :=\partial_a\partial_b\psi =\int_0^1 \operatorname{Tr}\!\left( \rho_\theta^u\widehat F_a \rho_\theta^{1-u}\widehat F_b \right)du.$$

It is real symmetric and positive semidefinite for Hermitian directions. It is positive definite after quotienting combinations of the $F_a$ that are scalar on the family. This canonical correlation is the Hessian induced by relative entropy Petz 1994, pp. 780–786.

For two points in the same family,

$$S(\rho_\theta\Vert\rho_{\theta'}) =\psi(\theta')-\psi(\theta) -(\theta'^a-\theta^a)\eta_a(\theta).$$

The argument order is part of the statement. Reversing it changes cubic and higher orders, though the same BKM quadratic form appears at coincidence.

Exponential and mixture affine structures

An exponential geodesic is affine in $\log\rho$ modulo its scalar normalization. A mixture geodesic is affine in the density matrix:

$$\rho_m(t)=(1-t)\rho_A+t\rho_B.$$

On the faithful state manifold the exponential and mixture connections are dual with respect to the BKM metric Hasegawa 1997, pp. 49–58. Among monotone metrics, constant multiples of BKM are singled out by this duality Grasselli and Streater 2001, pp. 173–182. “Dual” does not mean that the two curves coincide; it means their connections satisfy the metric duality relation.

Two smeared fermion modes: primal and dual paths

Take two orthonormal smooth wave packets $f_1,f_2$ in a finite spatial box with momentum cutoff $\Lambda$, and set $c_i=\psi(f_i)$. On the fixed one-particle sector, the bounded smeared bilinears

$$X=c_1^\dagger c_2+c_2^\dagger c_1, \qquad Z=c_1^\dagger c_1-c_2^\dagger c_2$$

act as $\sigma_x$ and $\sigma_z$. The two-parameter family

$$\rho(\theta_x,\theta_z) =\frac{e^{\theta_xX+\theta_zZ}} {2\cosh r}, \qquad r=\sqrt{\theta_x^2+\theta_z^2},$$

is faithful for every finite $r$, with identity-carrier potential $\psi=\log(2\cosh r)$ (equivalently, the $\rho_0=\mathbf1/2$ convention differs by an irrelevant constant) and Bloch expectation vector

$$\boldsymbol\eta =\frac{\tanh r}{r}(\theta_x,0,\theta_z).$$

Compare the endpoints $\rho_A=\rho(0,1)$ and $\rho_B=\rho(1,0)$. Their two affine paths are

$$\boldsymbol\theta_e(t)=(t,0,1-t), \qquad \boldsymbol\eta_m(t)=\tanh(1)(t,0,1-t), \qquad 0\le t\le1.$$

At $t=1/2$, the exponential path has $\boldsymbol\theta_e=(1/2,0,1/2)$, whereas the mixture path has

$$\boldsymbol\eta_m =\frac{\tanh1}{2}(1,0,1).$$

Converting the latter back to exponential coordinates gives a radial coordinate $r_m=\operatorname{artanh}(\tanh1/\sqrt2)$. The reproducible comparison is:

Midpoint	Exponential coordinate $(\theta_x,\theta_z)$	Bloch radius	Smallest eigenvalue
Exponential path	$(0.500000,0.500000)$	$0.608859$	$0.195570$
Mixture path	$(0.425735,0.425735)$	$0.538528$	$0.230736$

Both curves remain in the faithful $x$–$z$ plane, but their midpoint trace distance is

$$\frac12\lVert\rho_e(1/2)-\rho_m(1/2)\rVert_1 =\frac{0.608859-0.538528}{2} =0.0351655.$$

This finite-mode QFT application uses genuinely noncommuting smeared generators. Its controls are the box, cutoff $\Lambda$, fixed wave packets, fixed particle-number sector, parameter range, and smallest eigenvalue. A continuum comparison must embed the same $f_i$ and observables at successive cutoffs; changing the wave packets while changing $\Lambda$ confounds geometry with coarse graining.

What survives without a trace

A local QFT algebra is generally type III, so there is no density matrix or trace partition function intrinsic to that algebra. For a faithful normal reference state, bounded self-adjoint elements of the algebra can instead generate relative-Hamiltonian perturbations and intertwining cocycles Araki 1973, pp. 165–174 and 190–202. This is an algebraic exponential arc; it is not evidence that all normal states form one finite-dimensional manifold.

For an unbounded proposed generator $F$, one must establish at least:

• a dense domain and closability, followed by a specified self-adjoint realization or closed semibounded quadratic form;
• existence and normalizability of the perturbed state, or an algebraic perturbation theorem that replaces the trace exponential;
• faithfulness on the comparison algebra, or an explicit common support reduction;
• differentiability in a named topology and finiteness of the BKM tangent norm;
• regulator matching for smearing functions, counterterms, and the measured observables.

Closability is necessary but not sufficient: a closable symmetric operator may still have inequivalent self-adjoint extensions, and $e^{\log\rho_0+\theta F}$ may fail to be trace class. Conversely, an algebraic bounded perturbation can be meaningful even though no local density matrix exists.

Adversarial domain and support tests

The first failure injection deliberately chooses a nonclosable “sufficient statistic.” On $\ell^2(\mathbb N)$ let $\mathcal D(T)=c_{00}$ and

$$Tx=\left(\sum_{n=1}^\infty n x_n\right)e_1.$$

For $x^{(N)}=e_N/N$, $x^{(N)}\to0$ but $Tx^{(N)}=e_1$. The graph is not closable. Therefore $T$ cannot be promoted to a self-adjoint observable, functional calculus does not define $e^{\theta T}$ as a positive statistical family, and neither tangent nor BKM metric exists. Writing a formal partition function would hide the failure rather than cure it.

The second injection approaches the support boundary within a valid qubit family:

$$\rho_t=\frac{e^{tZ}}{2\cosh t}, \qquad p_{\min}(t)=\frac{1}{1+e^{2t}}\longrightarrow0.$$

Every finite $t$ is faithful, but the $t\to\infty$ limit is rank one. The natural coordinate diverges, $\psi''(t)=\operatorname{sech}^2t\to0$, and the dual metric $d t/d\eta=1/(1-\eta^2)$ diverges. The limiting pure state can be studied with a support-restricted or Bures construction, but it is not an interior point of this faithful BKM manifold.

Common pitfalls

Using $e^AB$ for the derivative of $e^{A+\lambda B}$. The Duhamel integral is essential when $A$ and $B$ do not commute.

Treating all continuum states as one exponential family. Normal-state sectors, supports, and unbounded perturbations impose genuine boundaries. A regulator family does not erase them.

Comparing unmatched cutoffs. The same sources, smearings, algebra embeddings, and observables must be held fixed before a change is attributed to information geometry.

Exercises

1. Derive the BKM Hessian

Starting from the Duhamel formula, show that $\partial_a\psi=\langle F_a\rangle$ and derive the stated integral for $\partial_a\partial_b\psi$. Explain why a scalar linear combination of the $F_a$ is a null direction.

Solution

Let $A(\theta)=\log\rho_0+\theta^aF_a$. Cyclicity of the trace gives

$$\partial_a\operatorname{Tr}e^A =\int_0^1\operatorname{Tr}(e^{uA}F_ae^{(1-u)A})du =\operatorname{Tr}(e^AF_a).$$

Dividing by $\operatorname{Tr}e^A$ yields $\partial_a\psi=\operatorname{Tr}(\rho F_a)=\eta_a$. Differentiating this expectation and substituting

$$\partial_b\rho=\int_0^1\rho^u(F_b-\eta_b\mathbf1)\rho^{1-u}du$$

gives the BKM integral. If $v^aF_a=c\mathbf1$, then $v^a\widehat F_a=0$, so $v^av^bg_{ab}=0$: that parameter changes only the discarded normalization scalar.

2. Reproduce the two midpoint geometries

For the two-mode family, compute the Bloch radii and smallest eigenvalues of the exponential and mixture midpoints. Verify the reported trace distance.

Solution

For the exponential midpoint, $r_e=1/\sqrt2$, hence $R_e=\tanh(1/\sqrt2)=0.608859$. For the mixture midpoint, averaging the endpoint Bloch vectors gives

$$R_m=\frac{\tanh1}{\sqrt2}=0.538528.$$

A qubit with Bloch radius $R$ has eigenvalues $(1\pm R)/2$, so the smaller eigenvalues are $0.195570$ and $0.230736$. The two midpoint Bloch vectors point in the same $x$–$z$ direction. Therefore

$$\frac12\lVert\rho_e-\rho_m\rVert_1 =\frac12\lvert R_e-R_m\rvert =0.0351655.$$

Finally, $r_m=\operatorname{artanh}R_m=0.602081$, so each of its two equal nonzero natural-coordinate components is $r_m/\sqrt2=0.425735$.

3. Diagnose both adversarial failures

Use the sequence $x^{(N)}=e_N/N$ to apply the graph criterion for closability of $T$. Then determine the asymptotic behavior of $p_{\min}(t)$ and of the dual metric for $\rho_t$.

Solution

An operator is closable only if $x_n\to0$ and $Tx_n\to y$ imply $y=0$. Here $\lVert x^{(N)}\rVert=1/N\to0$, while $Tx^{(N)}=e_1$. Thus $T$ is not closable and cannot have a closed, hence cannot have a self-adjoint, extension agreeing with it.

For the valid qubit family,

$$p_{\min}(t)=\frac{e^{-t}}{e^t+e^{-t}} =\frac{1}{1+e^{2t}}\sim e^{-2t}.$$

The expectation coordinate is $\eta=\tanh t$, so $d\eta/dt=1-\eta^2=\operatorname{sech}^2t$. Hence the dual coefficient $dt/d\eta=(1-\eta^2)^{-1}$ diverges as the faithful support is lost. The first example has no legitimate family at all; the second has a legitimate interior family but no faithful endpoint at $t=\infty$.

References

• Araki, Huzihiro. “Relative Hamiltonian for Faithful Normal States of a von Neumann Algebra.” Publications of the Research Institute for Mathematical Sciences 9 (1973): 165–209. DOI; Publisher page.
• Grasselli, Matheus R., and Raymond F. Streater. “On the Uniqueness of the Chentsov Metric in Quantum Information Geometry.” Infinite Dimensional Analysis, Quantum Probability and Related Topics 4 (2001): 173–182. DOI; Open PDF.
• Hasegawa, Hiroshi. “Exponential and Mixture Families in Quantum Statistics: Dual Structure and Unbiased Parameter Estimation.” Reports on Mathematical Physics 39 (1997): 49–68. DOI.
• Petz, Dénes. “Geometry of Canonical Correlation on the State Space of a Quantum System.” Journal of Mathematical Physics 35 (1994): 780–795. DOI.