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Quantum Entanglement as a Cohomological Obstruction

Kazuki Ikeda

TL;DR

This work reframes quantum entanglement as a cohomological obstruction to reconstructing a global state from locally compatible data, using a presheaf of states and Čech cohomology to capture gluing failures and non-uniqueness. It introduces a differential-geometric view via the amplitude bundle and an $A$-parallel entanglement witness, culminating in the Quantum Entanglement Index (QEI) which assigns an integer grade to entangled families. A bridge to geometric Langlands is built by pairing Čech data with Chern–Weil forms and linking Hecke modifications to entanglement shifts, defining a quantum entanglement refinement on automorphic categories. The paper also outlines practical implications for quantum many-body physics and gravity, including diagnostic tools (entanglement curvature and ν_ent) and potential experimental targets, and sketches how high-energy ideas like stress-tensor correlators relate to entanglement obstructions. Overall, the framework offers a topological and geometric language to quantify, classify, and detect entanglement in complex quantum systems, with concrete paths to experimental validation and cross-disciplinary connections.

Abstract

We recast quantum entanglement as a cohomological obstruction to reconstructing a global quantum state from locally compatible information. We address this by considering presheaf cohomologies of states and entanglement witnesses. Sheafification erases the global-from-local signature while leaving within-patch multipartite structure, captured by local entanglement groups introduced here. For smooth parameter families, the obstruction admits a differential-geometric representative obtained by pairing an appropriate witness field with the curvature of a natural unitary connection on the associated bundle of amplitudes. We also introduce a Quantum Entanglement Index (QEI) as an index-theoretic invariant of entangled states and explain its behavior. Finally, we outline a theoretical physics approach to probe these ideas in quantum many-body systems and suggest a possible entanglement-induced correction as an experimental target.

Quantum Entanglement as a Cohomological Obstruction

TL;DR

This work reframes quantum entanglement as a cohomological obstruction to reconstructing a global state from locally compatible data, using a presheaf of states and Čech cohomology to capture gluing failures and non-uniqueness. It introduces a differential-geometric view via the amplitude bundle and an -parallel entanglement witness, culminating in the Quantum Entanglement Index (QEI) which assigns an integer grade to entangled families. A bridge to geometric Langlands is built by pairing Čech data with Chern–Weil forms and linking Hecke modifications to entanglement shifts, defining a quantum entanglement refinement on automorphic categories. The paper also outlines practical implications for quantum many-body physics and gravity, including diagnostic tools (entanglement curvature and ν_ent) and potential experimental targets, and sketches how high-energy ideas like stress-tensor correlators relate to entanglement obstructions. Overall, the framework offers a topological and geometric language to quantify, classify, and detect entanglement in complex quantum systems, with concrete paths to experimental validation and cross-disciplinary connections.

Abstract

We recast quantum entanglement as a cohomological obstruction to reconstructing a global quantum state from locally compatible information. We address this by considering presheaf cohomologies of states and entanglement witnesses. Sheafification erases the global-from-local signature while leaving within-patch multipartite structure, captured by local entanglement groups introduced here. For smooth parameter families, the obstruction admits a differential-geometric representative obtained by pairing an appropriate witness field with the curvature of a natural unitary connection on the associated bundle of amplitudes. We also introduce a Quantum Entanglement Index (QEI) as an index-theoretic invariant of entangled states and explain its behavior. Finally, we outline a theoretical physics approach to probe these ideas in quantum many-body systems and suggest a possible entanglement-induced correction as an experimental target.

Paper Structure

This paper contains 16 sections, 16 theorems, 145 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Quantum entanglement from the viewpoint of topology
  3. Presheaf of quantum states
  4. Entanglement obstructions
  5. Feasibility by duality
  6. Obstructions in pure states
  7. Local entanglement
  8. Examples of local entanglement groups
  9. A differential geometric perspective of the obstruction
  10. Quantum entanglement index and quantum geometric Langlands correspondence
  11. Motivations
  12. Definition and basic properties
  13. Geometric Langlands correspondence with quantum entanglement
  14. Implications to quantum physics: from condensed matter and gravity
  15. Toward practical detection of entanglement via QEI in quantum many-body systems
  16. ...and 1 more sections

Key Result

Proposition 1.2

Let $\mathcal{U}=\{U_i\}_{i=1}^m$ be a partition of $I$. Given pure local states $\rho_i=|\psi_i\rangle\!\langle\psi_i|\in P(U_i)$, there exists a unique global pure state $\rho=|\Psi\rangle\!\langle\Psi|\in P(I)$ with $\rho|_{U_i}=\rho_i$ for all $i$, namely $|\Psi\rangle=\bigotimes_{i=1}^m |\psi_i

Figures (6)

  • Figure :
  • Figure :
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  • ...and 1 more figures

Theorems & Definitions (44)

  • Example 1.1
  • Proposition 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 34 more