Quantum Fisher Information as a Spectral Entanglement Witness

Overview

A recent inelastic-neutron-scattering experiment on the heavy-fermion strange metal $\mathrm{Ce}_3\mathrm{Pd}_{20}\mathrm{Si}_6$ reports a scale-free increase of the quantum Fisher information (QFI) density as the strange-metal state forms, witnessing high multipartite entanglement. This note asks whether the non-injective projection framework reproduces this signature, and isolates the exact structural condition that controls it.

Using linear response, the QFI density reduces to a weighted integral of the imaginary part of the dynamical susceptibility, whose infrared behaviour is fixed by the projective stability gap $\Delta_\Pi$ and the edge spectral weight of the projective stability operator $L_\Pi$. This yields a clean dichotomy: a finite gap forces the QFI density to saturate, whereas a gap that closes with edge spectral dimension $d_s > 2$ forces power-law growth $f_Q(T)\sim T^{1-a}$ with $a = d_s/2$.

Scope statement. This page is an entry point. The authoritative technical reference is the note linked above. The note tests whether the observed infrared type is compatible with the projective spectral data; it does not derive a microscopic Kondo-destruction model of the material.

Core results

Spectral dichotomy. Scale-free QFI growth is equivalent to two spectral statements about $L_\Pi$: the gap closes, and the edge density of states is anomalous with exponent $a > 1$ (otherwise growth is at most marginal).
Single sector is marginal. The winding sector $w=2$ reduces to the critical almost-Mathieu (Harper) operator at flux $2/q$; its gap closes as $\Delta_\Pi\sim q^{-1}$ but its edge spectral dimension is $d_s\to 2$, giving $a\to 1$ — the marginal boundary, hence only logarithmic growth.
Full substrate is power-law. The full Cayley substrate of $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$, of homogeneous growth dimension $D=4$, has edge spectral dimension $d_s\approx 4$, giving $a\approx 2$ and $f_Q(T)\sim T^{-1}$, a genuine power law. An exact block decomposition (one abelian torus band plus $q-1$ staggered Harper bands) reconstructs $d_s=4$ by band stacking.
Structural origin. The scale-free entanglement signature is a property of the full nilpotent substrate and not of any isolated winding sector — consistent with the measured entanglement being multipartite, of depth at least nine.
Thermal closure derived. The required closure $\Delta_\Pi(T)=\tfrac{d_s}{2}T$ follows from the framework's heat-kernel identification of the diffusion scale with inverse temperature; the trace-return plateau $u(t)\,t\to d_s/2$ is confirmed numerically (median $2.07, 2.04, 2.03$ at $q=61,101,151$).

Status of claims

The reduction of the QFI density to the projective spectral data is structural, conditional on a relaxational response form and on a flat matrix-element weight (recorded as inputs). The marginality of the $w=2$ sector and the power-law of the full substrate are numerical (exact diagonalisation and exact block assembly to $q=151$). The thermal closure $\Delta_\Pi(T)=\tfrac{d_s}{2}T$ is derived. The predicted exponent is of the right order but steeper than observed ($\alpha_{\mathrm{th}}=1$ versus $\alpha_{\mathrm{exp}}\approx 0.7$); the Born–Infeld bounded-response correction is shown numerically to soften it, with a size-stable crossing at a partial substrate saturation $\langle S\rangle\approx 0.21$, while the asymptotic exponent remains open.

A new spectral witness

The framework already contains two entanglement witnesses, neither of which is the QFI: the projection entropy $S_\Pi$, which bounds Bell–CHSH violations, and the effective entanglement observable $E_{\mathrm{eff}}(C)=\Delta_\Pi(C)(1-C^\nu)$. The latter is proportional to $\Delta_\Pi$ and therefore vanishes as the gap closes, whereas the QFI density diverges as the gap closes when $a>1$. The QFI is thus a genuinely new spectral witness in the framework, complementary to the entanglement sub-programme: it counts the multipartite correlation depth of the steady state, which grows as the projective stability gap closes.

Reproducibility

The five figures are produced by self-contained Python scripts (exact diagonalisation and exact block assembly): the marginal $w=2$ sector, the full-substrate edge density of states, the $q$-convergence and band stacking, the Planckian thermal closure, and the Born–Infeld softening across four system sizes. The thermal-closure plateau and the Born–Infeld crossing reproduce to the reported digits.

References

Beau, J. Quantum Fisher Information as a Spectral Entanglement Witness: the Projective Stability Gap and the Origin of Scale-Free Growth. Preprint, 2026. https://doi.org/10.5281/zenodo.21001565