Quantum Fisher Information and Bulk Symplectic Forms

Quantum Fisher information is a state-space metric; a bulk symplectic form is an antisymmetric form on classical solution space. Holography relates them only after the modular generator supplies a complex or Hamiltonian structure, turning the symplectic pairing into canonical energy. Gauge reduction, boundary counterterms, and the choice of information metric are part of the statement.

Required background. Quantum Fisher Information in QFT supplies the state-space Hessian. Canonical Energy and Second-Order Relative Entropy supplies the holographic quadratic identity.

Helpful background. Symplectic Forms, Hamiltonian Flows, and Poisson Brackets and Presymplectic Systems and the Covariant Phase-Space Ambiguity Map supply the geometric distinction. Bures, Kubo–Mori, and Monotone Metrics, Fidelity Susceptibility and Relevant Deformations, and Noncommutative Information Geometry and Exponential Families fix the metric choices and source families.

First application. Evaluate the Fisher metric for a one-parameter family of Euclidean-source states and reproduce the bulk symplectic form.

Which Fisher metric?

For a one-parameter state family $\rho(\lambda)$, the Hessian of relative entropy defines the Bogoliubov–Kubo–Mori form,

$$\mathcal I_{\mathrm{BKM}}(\delta\rho,\delta\rho) =\left.\frac{d^2}{d\lambda^2} S\bigl(\rho(\lambda)\Vert\rho(0)\bigr) \right|_{\lambda=0}.$$

The symmetric-logarithmic-derivative, Bures, and BKM metrics are different monotone metrics except in commuting or special pure-state limits. A holographic equality derived from relative entropy concerns the BKM Hessian unless another definition is explicitly established.

From Euclidean sources to Lorentzian data

Prepare a state with Euclidean sources $J^-(\tau,x)$ below the reflection slice and conjugate sources $J^+$. The quadratic normalization functional is built from the connected two-point function. Varying a holomorphic and an antiholomorphic source direction produces a Kähler form on the source manifold.

The bulk fields sourced by $\delta J^\pm$ solve the linear Euclidean equations. Continuing their Cauchy data to the Lorentzian slice gives perturbations $\delta_1\Phi$ and $\delta_2\Phi$. The renormalized bulk symplectic form is

$$\Omega_\Sigma(\delta_1\Phi,\delta_2\Phi) =\int_\Sigma\omega_{\mathrm{ren}} (\Phi;\delta_1\Phi,\delta_2\Phi).$$

With the source-space complex structure $J$, the positive metric is schematically

$$G(\delta\Phi,\delta\Phi) =\Omega_\Sigma(\delta\Phi,J\delta\Phi).$$

For a ball-region relative-entropy problem, $J$ is implemented by modular evolution and the right side becomes canonical energy. This is why the symmetric information metric can be represented through an antisymmetric symplectic current.

Gauge reduction and renormalization

Before quotienting, the gravitational symplectic form is presymplectic: infinitesimal diffeomorphisms are null only when they vanish appropriately at every boundary. A transformation that moves the asymptotic sources or extremal surface carries a charge and cannot be discarded.

Counterterms add finite boundary contributions to $\omega_{\mathrm{ren}}$. The same scheme must be used in the boundary generating functional. A source contact term can shift local pieces of the information metric; separated-source or properly subtracted quantities are the invariant comparison.

Positivity and degeneracy checks

The boundary metric is nonnegative by information theory. The bulk expression must therefore be nonnegative on perturbations that correspond to physical state variations. Negative norm signals an instability, a wrong contour, an unremoved ghost, or a mismatch in boundary conditions.

Zero norm can mean a gauge direction, an operator null direction, or a source variation that does not change the restricted state. These possibilities are separated by checking boundary charges and the correlator kernel.

Adversarial controls

Switch metrics silently. Replacing the BKM Hessian by the Bures metric changes the modular kernel. Agreement in a commuting toy model does not justify the substitution in QFT.

Keep gauge volume. A nontrivial norm for a compactly supported pure diffeomorphism exposes a missing constraint or boundary term.

Compare bare forms. An unrenormalized bulk symplectic flux cannot equal a finite CFT information quantity.

Evidence ceiling

For Euclidean-source state families in a controlled code sector, the renormalized bulk symplectic form combined with the relevant complex or modular structure reproduces the boundary information metric. The result does not identify all monotone metrics, prove nonperturbative bulk phase space, or make positivity global beyond the state family.

The equality between the relative-entropy Hessian and canonical energy in the specified semiclassical code sector is the result of Lashkari and Van Raamsdonk 2016; it is not a claim that every quantum Fisher metric equals one bulk symplectic form.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• de Boer, Jan, Michal P. Heller, Robert C. Myers, and Yasha Neiman. “Holographic Kähler and Information Metrics.” Journal of High Energy Physics 2016, 149 (2016). DOI; arXiv:1509.00113.
• Lashkari, Nima, and Mark Van Raamsdonk. “Canonical Energy Is Quantum Fisher Information.” Journal of High Energy Physics 2016, 153 (2016). DOI; arXiv:1508.00897.