Quantum Entanglement, Stratified Spaces, and Topological Matter: Towards an Entanglement-Sensitive Langlands Correspondence

TL;DR

The paper investigates how quantum entanglement functions as a cohomological obstruction to reconstructing a global quantum state from locally compatible data, using a stratified parameter-space framework and the Haldane model as a concrete testbed. It develops an entanglement witness filtering scheme that yields a curvature-weighted coherence and exact lattice identities linking Chern numbers to sector-resolved responses, while extending the picture to multi-orbital settings with matrix-valued coherence and Levi-type classifications. A key technical advance is the introduction of the $S$-filtered quantum geometric tensor and quantum Fisher information, with rigorous bounds showing how entanglement channels bound metrological resources and how the filtered responses peak near Dirac-like singularities. The results connect a topological Langlands-type duality to physically measurable jumps across stratification walls, offering a practical tomography route for entanglement-encoded topological data and a framework extendable to mixed states, non-Abelian degeneracies, and broader topological phases.

Abstract

Recently, quantum entanglement has been presented as a cohomological obstruction to reconstructing a global quantum state from locally compatible information, where sheafification provides a functor that is forgetful with regards to global-from-local signatures while acting faithfully with respect to within-patch multipartite structures. Nontrivial connections to Hecke modifications and the geometric Langlands program are explored in the process. The aim of this work is to validate and extend a number of the claims made in [arXiv:2511.04326] through both theoretical analysis and numerical simulations, employing concrete perspectives from condensed matter physics.

Figures (1)

• Figure 1: (a) FHS curvature $F(\mathbf k)$; its sum yields the Chern number $\mu=\tfrac{1}{2\pi}\sum_{\mathbf k}F(\mathbf k)$. (b) $\alpha(\mathbf k;\theta)=\tfrac{1}{2}+\Re\!(e^{i\theta}v_A v_B^{\ast})$. (c) Graded density $\langle S\rangle F/(2\pi)$ with $\langle S\rangle=1-2\alpha$; its sum is the graded response $\nu(\theta)$. (d) $\mu(M)$, $\ \nu_{\pm}(M)$ and $\nu(M)=\nu_+(M)-\nu_-(M)$, confirming the discrete identities $\mu=\nu_++\nu_-$ and $\nu=\nu_+-\nu_-$. The quantized jumps of these responses at the critical values of $M$ coincide with the crossings of the gap-closing stratum $\Sigma$ and realize the Hecke jumps $\Delta \mathrm{Ind}_S$ along this path. (e) Residuals $r_\mu=\mu-(\nu_++\nu_-)$ and $r_\nu=\nu-(\nu_+-\nu_-)$, confirmed to be negligibly small, on the order of $10^{-15}$. (f) Tomography (reconstruction of $J_F$): exact $\nu(\theta)=-2\,\Re\!(e^{i\theta}J_F^{AB})$ against the two‑witness reconstruction from $\theta=0,\pi/2$, where $J_F^{AB}=\tfrac{1}{2\pi}\sum_{\mathbf k}F\,v_A v_B^{\ast}$. (g) Multi‑orbital samples: $\nu_-(x,y;\theta)=\tfrac{\mu}{2}+\Re\!(e^{i\theta}x^\dagger J_F y)$ with $J_F=\tfrac{1}{2\pi}\sum_{\mathbf k}F\,a\,b^\dagger$ for the embedding $a=v_A x$, $b=v_B y$. (h) $S$-filtered QFI inequality: $\mathcal{F}^{\mathrm Q}=4g_{ii}$ and $\mathcal{F}^{\mathrm Q,(S)}\le 4\,\|P^\perp S'P^\perp\|\,g_{ii}\le\mathcal{F}^{\mathrm Q}$. Here, all random scatter points lie below the line $y=x$.