Quantum Fisher Information in QFT

Quantum Fisher information quantifies the best local statistical distinguishability available from a parameterized quantum state. For the symmetric logarithmic derivative it equals the maximum classical Fisher information over measurements, but only after the state family, support, normalization, and allowed measurement class are fixed. In QFT, sharp local tangents often make it infinite unless smearing or energy constraints are imposed.

Required background. State perturbations and relative-entropy susceptibility supply normalized tangents and a comparison with the Kubo–Mori Hessian.

Helpful background. Operational distinguishability bounds supply the measurement interpretation.

Symmetric logarithmic derivative

Let $\rho_\lambda$ be a differentiable density-operator family and write $X=\partial_\lambda\rho_\lambda$ . The symmetric logarithmic derivative $L_\lambda$ is defined on the support by

X=\frac12(\rho L+L\rho).

The SLD quantum Fisher information is

F_Q(\lambda)=\operatorname{Tr}(\rho L^2) =\operatorname{Tr}(XL).

For a faithful $\rho=\sum_n p_n\lvert n\rangle\langle n\rvert$ ,

L_{mn}=\frac{2X_{mn}}{p_m+p_n}, \qquad F_Q=\sum_{m,n}\frac{2\lvert X_{mn}\rvert^2}{p_m+p_n}.

Terms with $p_m+p_n=0$ lie outside the support and require a support-changing prescription. If an eigenvalue approaches zero while $X$ connects to that direction, $F_Q$ can diverge.

For a normalized pure state $\lvert\psi_\lambda\rangle$ ,

F_Q =4\left( \langle\partial_\lambda\psi\mid\partial_\lambda\psi\rangle -\lvert\langle\psi\mid\partial_\lambda\psi\rangle\rvert^2 \right).

The subtraction removes the phase direction and makes the expression gauge invariant.

The structural map places Quantum Fisher Information in QFT along the perturbative sequence from normalized state or shape variations to information metrics and modular transport.

A normalized family yields the fixed-reference first law and a positive quadratic response, which branches into state susceptibility, shape kernels, metric choices, and modular holonomy. — 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.

Measurement interpretation

A POVM $\{M_x\}$ produces probabilities $p(x\mid\lambda)=\operatorname{Tr}(\rho_\lambda M_x)$ and classical Fisher information

F_{\{M_x\}} =\sum_x p(x\mid\lambda) \left[\partial_\lambda\log p(x\mid\lambda)\right]^2.

Braunstein and Caves 1994, pp. 3440–3442 showed

F_{\{M_x\}}\le F_Q,

with equality attainable locally by a measurement adapted to the SLD under the usual regularity conditions. “Attainable locally” matters: the optimal measurement can depend on the unknown parameter, may be collective, and may not belong to an experimentally allowed local algebra.

For $\nu$ independent repetitions, the quantum Cramér–Rao inequality reads

\operatorname{Var}(\widehat\lambda) \ge\frac{1}{\nu F_Q}

for a locally unbiased estimator under standard identifiability assumptions. Biased estimators, nuisance parameters, finite samples, and correlated field measurements require modified bounds.

Comparison with relative-entropy geometry

The Kubo–Mori susceptibility has kernel

c_{\rm KM}(p_m,p_n) =\frac{\log p_m-\log p_n}{p_m-p_n},

whereas SLD Fisher information has

c_{\rm SLD}(p_m,p_n)=\frac{2}{p_m+p_n}.

They coincide for commuting tangents, including diagonal probability variations. For coherent directions they differ because the logarithmic and arithmetic means differ. With the normalizations above,

F_Q(X,X)\le \chi_{\rm KM}(X,X)

for a faithful finite state. Therefore a relative-entropy Hessian should not be labeled SLD Fisher information unless the family is commuting or the conversion has been proved.

Both metrics contract under completely positive trace-preserving maps. Operationally, discarding observables cannot increase the best local distinguishability.

QFT tangents and constraints

In continuum QFT, a family can be specified as normal states $\omega_\lambda$ on a fixed local algebra. An SLD operator need not exist as a bounded element of that algebra even when a quadratic form exists. Useful controlled settings include:

a regulated lattice or mode algebra followed by convergence of the metric;
bounded source operators in the local algebra;
spacetime-smoothed fields with a common energy domain;
energy-constrained Fisher information for a specified class of states and measurements.

An unsmeared local unitary can excite arbitrarily high frequencies and make $F_Q$ divergent. Such a divergence may reflect genuine ultraviolet distinguishability rather than a calculational error, but it does not define a finite observable. State the cutoff scaling and the operational resolution.

If measurements are restricted to a subalgebra $\mathcal N\subset\mathcal A$ , the relevant Fisher information is that of the restricted state family, not the global value. Monotonicity guarantees it is no larger.

A unitary-family check

For $\rho_\lambda=e^{-i\lambda G}\rho e^{i\lambda G}$ ,

X_{mn}=-i(p_n-p_m)G_{mn},

and hence

F_Q =\sum_{m,n} \frac{2(p_m-p_n)^2}{p_m+p_n} \lvert G_{mn}\rvert^2.

For a pure state this reduces to $4\,\operatorname{Var}_\psi(G)$ . Components of $G$ commuting with $\rho$ vanish, as they must. This example is a useful check before applying a field-theory regulator.

Common pitfalls

Equating Fisher information with one chosen measurement’s Fisher information. The SLD value is an optimization. State the allowed POVMs and whether the optimum is achievable.

Ignoring support motion. The SLD inverse uses $p_m+p_n$ . Tangents entering a zero-probability sector can be singular or only one-sided.

Calling every positive modular Hessian Fisher information. Specify SLD, right logarithmic derivative, or Kubo–Mori geometry; they are inequivalent for noncommuting families.

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

A reported susceptibility must pass separate checks for normalized families, common tangent domains, contact and regulator terms, and an explicit metric and analytic strip before receiving a physical interpretation. — 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

Braunstein, Samuel L., and Carlton M. Caves. “Statistical Distance and the Geometry of Quantum States.” Physical Review Letters 72 (1994): 3439–3443. DOI.