Modular Response and Quantum Information Geometry

Modular response theory studies how entropy, relative entropy, modular generators, and distinguishability change along a controlled family of states or regions. The first derivative gives the entanglement first law at fixed algebra and reference state; the first nonvanishing relative-entropy term is quadratic and defines a positive susceptibility. For noncommuting families, however, “the information metric” is not unique: Bures, symmetric-logarithmic-derivative, Kubo–Mori, fidelity, and modular-response geometries weight operator directions differently.

Helpful background. Modular Hamiltonian definitions fix domains and conventions, relative entropy in QFT supplies the finite comparison, and out-of-time-order correlators clarify what additional assumptions are needed before calling analytic growth “chaos.”

Enter a differentiable family

Let $\rho_\lambda$ be a normalized faithful family on one fixed regulated algebra, with $\rho_0=\sigma$ and $\operatorname{Tr}\dot\rho_0=0$. Writing $K_0=-\log\sigma$, differentiation gives

$$\left.\frac{d}{d\lambda}S(\rho_\lambda)\right|_0 =\operatorname{Tr}(\dot\rho_0K_0) =\left.\frac{d}{d\lambda}\langle K_0\rangle_{\rho_\lambda}\right|_0.$$

Equivalently, the linear term in $S(\rho_\lambda\Vert\sigma)$ vanishes. The relative-entropy derivation used in QFT is reviewed in Blanco, Casini, Hung, and Myers 2013, §2. Positivity then begins at quadratic order:

$$S(\rho_\lambda\Vert\sigma) =\frac{\lambda^2}{2}\, \chi_{\rm rel}(\dot\rho_0,\dot\rho_0) +O(\lambda^3).$$

In continuum QFT, the separate entropy and modular-energy variations may be divergent while their relative-entropy combination is finite. The comparison algebra, smearing, state family, and regulator-removal order must remain fixed.

The diagram shows the common perturbative spine and the points where state, shape, metric, and transport questions branch apart.

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.

Begin with the entanglement first law and quadratic correction, then formulate state susceptibility on a controlled tangent domain. Region changes require separate shape-deformation theory and carefully ordered stress-tensor response kernels.

The metric sequence starts with quantum Fisher information, compares Bures, Kubo–Mori, and monotone metrics, and treats fidelity susceptibility near relevant deformations. The final pages develop modular Berry holonomy, conditional modular analyticity bounds, noncommutative exponential families, and a practical contact-term and error budget.

Independent validity gates

A response coefficient can fail for several unrelated reasons. Normalization may be wrong; the tangent may not lie in the metric domain; moving a region may introduce contact or boundary terms; two authors may use inequivalent monotone metrics; or an analytic continuation may lack the boundedness required for a growth estimate. Passing one check does not repair another.

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.

The safest workflow fixes the family and algebra first, computes a positive finite regulated quadratic form second, and only then asks which experimental or computational proxy represents it. State and region variations should be evaluated separately before any mixed derivative is interpreted.

Comparison of information geometries

Normalizations vary across fields. The table uses $F_Q=\operatorname{Tr}(\rho L^2)$ for the symmetric logarithmic derivative $L$, so the infinitesimal Bures line element is $ds_B^2=F_Q\,d\lambda^2/4$. The estimation normalization follows Braunstein and Caves 1994, pp. 3440–3442, while the monotone-metric classification follows Petz 1996, pp. 87–90. Every row assumes normalized tangents.

Information metrics and modular responses for controlled quantum-state families.

Construction	Family and tangent	Normalization or kernel	Positivity and monotonicity	Continuum control	Operational or computational proxy
SLD quantum Fisher information	Faithful density operators with tangent defined by one-half of ρL + Lρ	Fisher information is Tr ρL²	Positive and contractive under channels; bounds locally unbiased estimation	May diverge for sharp local perturbations; specify smearing, energy constraint, and support	Optimal measurement Fisher information and the quantum Cramér–Rao bound
Bures metric	Mixed-state family compared through Uhlmann fidelity	One quarter of SLD Fisher information in the convention used here	Positive and monotone; minimal among standard monotone metrics with matched normalization	Fidelity of continuum reductions needs a common algebra and regulator-safe limit	State overlap, purification distance, and interferometric protocols
Bogoliubov–Kubo–Mori metric	Faithful noncommuting family, naturally in exponential coordinates	Hessian of relative entropy or log partition function; logarithmic-mean kernel	Positive and monotone; differs from Bures away from commuting directions	Relative quantities can cancel UV terms, but tangent-domain and contact checks remain	Static susceptibility and imaginary-time connected correlators
Fidelity susceptibility	Ground-state, thermal, or reduced-state family under a parameter deformation	Quadratic loss of fidelity; equals a fixed multiple of Fisher information only after conventions and state class are matched	Nonnegative, but extensivity and critical scaling are model dependent	Often contains volume and cutoff terms; universal parts require subtraction or scaling analysis	Overlap derivatives, spectral sums, and integrated deformation correlators
Relative-entropy modular response	Fixed algebra and reference state with a differentiable normal tangent	Second variation of relative entropy, equivalent to Kubo–Mori geometry in the faithful finite setting	Positive by relative-entropy positivity and monotone under restriction or channels	Frequently the best continuum object after state-independent divergences cancel	Modular-flow kernels, stress-tensor response, and regulated correlator differences

What the chapter establishes

The chapter establishes exact differential identities in faithful finite settings and explains the extra hypotheses needed for QFT limits. It does not identify one metric as universally measurable, equate every susceptibility with a phase-transition observable, or infer physical chaos from modular KMS analyticity alone. Modular Berry transport is defined only after projecting zero modes and controlling degeneracies; growth bounds are conditional on a normalized analytic function with an explicit strip bound.

References

• Blanco, David D., Horacio Casini, Ling-Yan Hung, and Robert C. Myers. “Relative Entropy and Holography.” Journal of High Energy Physics 2013, no. 8 (2013): 060. DOI; arXiv.
• Braunstein, Samuel L., and Carlton M. Caves. “Statistical Distance and the Geometry of Quantum States.” Physical Review Letters 72 (1994): 3439–3443. DOI.
• Petz, Dénes. “Monotone Metrics on Matrix Spaces.” Linear Algebra and its Applications 244 (1996): 81–96. DOI.