State Perturbations and Relative-Entropy Susceptibility

A relative-entropy susceptibility is the Hessian of relative entropy along a differentiable state family on one fixed algebra. In a faithful finite system it is the Bogoliubov–Kubo–Mori quadratic form. In QFT it is meaningful only for tangent directions whose relative modular response is finite after smearing, energy restriction, or regulator removal.

Required background. The first law and quadratic correction supply the Hessian. Helpful background. Araki relative entropy supplies the algebraic continuum definition.

Tangents to normalized state space

Let $\omega_\lambda$ be a differentiable family of normal states on a fixed von Neumann algebra $\mathcal A$, with reference $\omega_0=\omega$. Its tangent is the normal linear functional

$$\dot\omega(A)= \left.\frac{d}{d\lambda}\omega_\lambda(A)\right|_0, \qquad \dot\omega(\mathbf1)=0.$$

In a matrix representation, $\dot\omega(A)=\operatorname{Tr}(XA)$ with $X=X^\dagger$ and $\operatorname{Tr}X=0$. Not every traceless operator is an admissible two-sided tangent at a nonfaithful state: positivity of $\rho_\lambda$ constrains components that enter or leave the support.

The relative-entropy susceptibility, obtained by differentiating the Araki divergence of Araki 1976, pp. 809–817, is

$$\chi_\omega(\dot\omega,\dot\omega) =\left. \frac{d^2}{d\lambda^2} S(\omega_\lambda\Vert\omega) \right|_{\lambda=0},$$

when the derivative exists and is finite. For two tangent directions $X,Y$, polarization defines a symmetric bilinear form.

The structural map places State Perturbations and Relative-Entropy Susceptibility along the perturbative sequence from normalized state or shape variations to information metrics and modular transport.

At fixed comparison algebra, normalization removes the linear term in relative entropy. The second response supports several inequivalent constructions: state and shape tangents, stress-tensor kernels, monotone metrics, and zero-mode-projected transport. The diagram is schematic and not to scale.

Finite faithful formula

For $\rho>0$ and $\rho_\lambda=\rho+\lambda X+O(\lambda^2)$,

$$\chi_\rho(X,Y) =\operatorname{Tr}\!\left[ X\int_0^\infty d\beta\, (\rho+\beta)^{-1}Y(\rho+\beta)^{-1} \right].$$

In the eigenbasis of $\rho$ this becomes

$$\chi_\rho(X,Y) =\sum_{m,n} \frac{\log p_m-\log p_n}{p_m-p_n} X_{nm}Y_{mn}.$$

The kernel is the reciprocal logarithmic mean. It treats diagonal and coherent tangents differently, and it is not generally equal to the symmetric-logarithmic-derivative Fisher metric. They coincide in a classical commuting family only after matching normalization.

The map inverse to the Hessian kernel is

$$\Omega_\rho(A) =\int_0^1 du\;\rho^u A\rho^{1-u}.$$

If $X=\Omega_\rho(L_{\rm KM})$, then $\chi_\rho(X,X)=\operatorname{Tr}[X L_{\rm KM}]$. This Kubo–Mori logarithmic derivative is distinct from the symmetric logarithmic derivative.

A unitary perturbation check

Let $\rho_\lambda=e^{-i\lambda G}\rho e^{i\lambda G}$ with bounded Hermitian $G$. Then

$$X=-i[G,\rho], \qquad X_{mn}=-i(p_n-p_m)G_{mn}.$$

Substitution gives

$$\chi_\rho =\sum_{m,n} (p_m-p_n)(\log p_m-\log p_n) \lvert G_{mn}\rvert^2\ge0.$$

Directions generated by operators commuting with $\rho$ vanish because they do not change the state. This provides a strong check on signs, double counting, and zero modes.

For a source-prepared family, the tangent is often written

$$\rho_\lambda =\frac{e^{-\lambda V/2}\rho\,e^{-\lambda V/2}} {\operatorname{Tr}(\rho e^{-\lambda V})}.$$

Differentiating the denominator subtracts $\langle V\rangle_\rho$ and enforces $\operatorname{Tr}X=0$. Omitting this connected subtraction creates a false linear term and contaminates the susceptibility.

Continuum QFT domain

In a local QFT, an unsmeared field insertion may produce an infinite quadratic form. A controlled tangent can be specified by:

• a bounded element of the local algebra;
• a spacetime-smeared source with support separated from singular boundaries;
• an energy-bounded vector or form domain;
• or a regulated family whose relative entropy converges monotonically under algebra restriction.

The separate entropy Hessian and modular-energy Hessian can contain UV divergences that cancel in relative entropy. This cancellation should be demonstrated at matched regulator, region, and source rather than assumed from power counting.

For perturbations created by Euclidean sources, susceptibility becomes an integrated connected two-point function with a modular or imaginary-time kernel. Coincident insertions require local counterterms. If the source moves the entangling surface or regulator support, the calculation is no longer a pure state tangent.

Monotonicity and coarse graining

Relative entropy decreases under a quantum channel or restriction to a subalgebra. Differentiating at the reference gives contraction of its Hessian:

$$\chi_{\Phi(\rho)} \bigl(\Phi(X),\Phi(X)\bigr) \le \chi_\rho(X,X)$$

for channels and tangents within the differentiability domain. In QFT, isotony therefore makes susceptibility nondecreasing as the observable algebra is enlarged, provided the same pair of states is restricted consistently.

This is a distinguishability statement. It does not imply that a particular detector or local operator attains the full susceptibility; operational accessibility requires an allowed measurement class.

Common pitfalls

Calling any second derivative a susceptibility. State the divergence, reference, fixed algebra, and tangent normalization. Bures and Kubo–Mori Hessians differ for noncommuting directions.

Forgetting the normalization derivative. Source-prepared families require connected subtraction. Otherwise $\dot\omega(\mathbf1)\ne0$.

Removing the UV cutoff separately in two divergent terms. Form relative entropy or its matched quadratic combination first, then take the common limit.

Before interpreting this response coefficient, use the validity map to check normalization, tangent domains, contact and regulator terms, metric choice, and analyticity independently.

Normalization, domain, contact-term, metric, and analyticity checks are independent. A failed gate narrows or withdraws the physical claim; it is not a change of notation. The decision map is schematic and not to scale.

References

• Araki, Huzihiro. “Relative Entropy of States of von Neumann Algebras.” Publications of the Research Institute for Mathematical Sciences 11 (1976): 809–833. DOI.

Further reading

• Petz, Dénes. “Monotone Metrics on Matrix Spaces.” Linear Algebra and its Applications 244 (1996): 81–96. DOI.