Modular Response and Quantum Information Geometry
Modular response studies how information changes under a controlled variation. Varying a state, moving a region, changing couplings, and transporting eigenspaces define different derivatives even when their formulas look similar.
Enter this chapter
Section titled “Enter this chapter”Begin with First Law of Entanglement and Quadratic Corrections to specify a differentiable family and what remains fixed.
Helpful background. Modular Hamiltonian definitions fix domains and additive conventions, relative entropy in QFT supplies the finite comparison, and out-of-time-order correlators clarify why analytic growth is not by itself a diagnosis of physical chaos.
Topics
Section titled “Topics”- First Law of Entanglement and Quadratic Corrections — Derives the entanglement first law at a fixed reference state.
- State Perturbations and Relative-Entropy Susceptibility — Interprets second-order relative entropy as a susceptibility.
- Shape Deformations and Modular Perturbation Theory — Identifies the algebra when a region boundary moves.
- Modular Response Kernels and Stress-Tensor Insertions — Distinguishes stress response from modular and information kernels.
- Quantum Fisher Information in QFT — Defines quantum Fisher information for parameter estimation.
- Bures, Kubo–Mori, and Monotone Metrics — Compares inequivalent quantum information metrics.
- Fidelity Susceptibility and Relevant Deformations — Connects ground-state overlap to deformation susceptibility.
- Modular Berry Transport and Holonomy — Defines modular transport after resolving zero-mode freedom.
- Modular Analyticity and Chaos Bounds — Applies analyticity bounds only in their stated domains.
- Noncommutative Information Geometry and Exponential Families — Studies information geometry on noncommutative state families.
- Contact Terms, Operator Domains, and Perturbative Error Budgets — Tracks contacts, differentiability, and errors in response calculations.
How the ideas fit together
Section titled “How the ideas fit together”At first order around a fixed reference, the variation of entropy equals the variation of modular energy. The relative entropy begins at second order and measures distinguishability along the chosen state family Blanco, Casini, Hung, and Myers 2013, § 2.
This does not make every quantum information metric identical. Different metrics correspond to different constructions or estimation tasks. A shape deformation further changes the observable algebra, so a comparison map and contact terms are necessary. Treating a moving-region derivative as an ordinary state derivative can omit precisely the boundary response being studied.
The comparison below separates the assumptions and tests needed for these different conclusions.
Comparison of information geometries
Section titled “Comparison of information geometries”| Construction | Family and tangent | Normalization or kernel | Positivity and monotonicity | Continuum control | Operational or computational proxy |
|---|---|---|---|---|---|
| SLD quantum Fisher information | Faithful density operators, or a fixed-support extension, with tangent equal to one-half of ρL + Lρ | Fisher information is Tr ρL²; the corresponding Bures line element is one quarter of this value | Positive and contractive under quantum channels; gives the smallest normalized monotone metric | Sharp local parameter directions can diverge; specify smearing, support, and energy control | Maximum classical Fisher information over measurements and the quantum Cramér–Rao bound |
| Bures metric | State family compared through Uhlmann fidelity, including a controlled fixed-support boundary limit | One quarter of SLD Fisher information when Bures distance is defined by 2(1 − root fidelity) | Positive and channel contractive; four times the geometric Bures metric is the minimal Petz-normalized monotone metric | Continuum fidelity requires one comparison algebra and a regulator-safe limit | Purification distance, optimal purification overlap (or pure-state overlap), and locally optimal estimation |
| Bogoliubov–Kubo–Mori metric | Faithful noncommuting family, naturally described in exponential coordinates | Hessian of relative entropy or log partition function with the logarithmic-mean kernel | Positive and channel contractive; equals the classical Fisher metric on commuting directions but generally exceeds SLD with matched normalization | Relative quantities can cancel state-independent UV terms, but tangent-domain and contact checks remain | Static susceptibility, imaginary-time connected correlators, and relative-entropy response |
| Fidelity susceptibility | Ground-state, thermal, or reduced-state family under a named parameter deformation | Quadratic fidelity loss; its factor relative to Fisher information depends on rooted-versus-squared fidelity convention | Nonnegative, while extensivity, singularity, and critical scaling are model and direction dependent | Often contains volume and cutoff terms; universal parts require subtraction or finite-size scaling | Overlap derivatives, spectral sums, and integrated deformation correlators |
| Relative-entropy modular response | Fixed algebra and reference state with a differentiable normalized tangent | Second variation of relative entropy; the Kubo–Mori metric in the faithful regulated setting | Positive and monotone under restriction or quantum channels | Often the most robust continuum comparison after matched UV terms cancel | Modular-flow kernels, stress-tensor response, and regulated correlator differences |
Where to go next
Section titled “Where to go next”Continue to Information Across Scales and Renormalization to separate region size, regulator, coupling, and renormalization scales.