Presheaf Cohomologies of States

• Presheaf cohomologies are a mathematical framework that reconceptualizes quantum entanglement as a global gluing obstruction of locally defined quantum states.
• The methodology integrates cosimplicial differentials, witness fields, and the Quantum Entanglement Index (QEI) to bridge algebraic, geometric, and operational descriptions.
• Key implications include unifying discrete entanglement measures with topological invariants and offering experimental pathways through entanglement-filtered diagnostics.

Presheaf cohomologies of states provide a rigorous mathematical framework for capturing the global signature of quantum entanglement as a topological or cohomological obstruction. This perspective unifies the algebraic, geometric, and operational facets of entanglement. Instead of viewing entanglement purely as a property of individual states or tensors, it is recast as a failure in patching together local quantum data—encoded as sections of a presheaf or sheaf—into a global state, quantifiable via Čech or sheaf cohomology. The primary technical advance is the identification of an index-theoretic invariant—termed the Quantum Entanglement Index (QEI)—that captures this obstruction and admits both differential-geometric and index-theoretical interpretations (Ikeda, 6 Nov 2025). This cohomological framework integrates and extends ideas from quantum information, geometric topology, and gauge theory.

1. Presheaf Cohomology and Global State Reconstruction

A quantum system composed of a set of subsystems $I$ is associated with a collection of density matrices over each subsystem and their unions. This structure can be formalized as a presheaf $U \mapsto S(U) \subset \mathrm{End}(H_U)$ over the poset of subsystem subsets, with restriction maps implemented by partial traces. The presheaf is generally not a sheaf: given locally compatible marginal states on overlapping subsystems, there may be no global quantum state that restricts to these marginals. The obstruction to global gluing is measured by certain Čech cohomology classes $H^q(U, S(\cdot))$.

For a general system, the group $$R^0(U) = \ker[\mathrm{End}(H_I) \to H^0(U,\mathrm{End}(H_\bullet))]$$ quantifies the kernel of the reconstruction map; i.e., the degree to which local data fails to determine a unique global state. More refined "local entanglement groups" $E^q(U) = H^q(C^{0,\bullet}(U), \delta_E)$ can be constructed using ancilla-augmented cosimplicial differentials, which encode multipartite structure and ancilla manipulations.

2. Witness Fields and Differential-Geometric Representatives

When the family of quantum states depends smoothly on external parameters (e.g., spatial position or external fields), this local-to-global obstruction acquires a geometric representative. States $\rho(x)$ parametrized by $x \in X$ (with $X$ a smooth manifold) naturally give rise to a vector bundle $E \to X$ of amplitudes and a unitary connection $A$ (specifically, the Uhlmann connection maximizing fidelity).

A "witness field" $W \in \Gamma(\mathrm{End} E)$ is then selected, subject to parallel transport ($D_A W = 0$). The spectral decomposition of $W$ yields a sign operator $S = \mathrm{sgn}\,W$, breaking $E$ into positive, negative, and zero-eigenvalue subbundles. The main result (Ikeda, 6 Nov 2025) establishes that the cohomological (Čech) obstruction paired with $W$ corresponds, via the Čech–de Rham isomorphism, to the closed form $(2\pi i)^{-k} \mathrm{Tr}(W F_A^k)$ and the associated Chern–Weil representative in de Rham cohomology.

3. Quantum Entanglement Index: Index-Theoretic Invariant

The central invariant is the Quantum Entanglement Index (QEI), defined as the $S$-graded analytic index of the chiral Dirac operator twisted by the vector bundle $E$: $$\mathrm{QEI} = \mathrm{Ind}_S(D_X \otimes E) = \langle \hat{A}(TX) \wedge \mathrm{Tr}[S \exp(F_A/2\pi i)], [X] \rangle \in \mathbb{Z}$$ Here, $\hat{A}(TX)$ is the A-roof genus of the tangent bundle, $F_A$ the curvature of $A$, and the sign operator $S$ grades the index, associating positive/negative contributions depending on the sign of the witness field's eigenvalues.

The construction satisfies:

• Integrality: The index is always an integer.
• Deformation invariance: Under smooth deformations preserving the parallelism of $W$, QEI is unchanged.
• Additivity: For direct sum bundles and direct sum witness fields, the QEI adds accordingly.
• Local unitary/LOCC invariance: Since the obstruction is defined globally on the presheaf and the witness field respects the connection, operations that act as local unitaries/LOCC preserve the QEI.

4. Example Computations and Physical Interpretation

Physical and toy examples illustrate the formalism:

• Trivial product states: When the state bundle is trivial and the curvature vanishes, all cohomological and index-theoretic invariants vanish (QEI = 0), reflecting the absence of entanglement.
• Bloch sphere family: For pure qubit state families parametrized over $S^2$, the QEI reduces to the Chern number of the Hopf bundle, but does not capture true entanglement (single-subsystem case).
• GHZ-type families: For parameter spaces encoding genuine multipartite entanglement (e.g., GHZ phase spaces), QEI becomes nonzero and measures the net “charge” of entanglement sector transfer across the parameter space, interpreted as band crossings in the amplitude bundle induced by entanglement transitions.

A general recipe is as follows: parametrize the state family $\rho(x)$, construct local amplitude frames $W_i$, calculate the Uhlmann connection $A_i$ and its curvature $F_i$, choose a parallel witness field $W$, and compute $\int_X \mathrm{Tr}\, (S F_A)/(2\pi i)$ over the parameter space, yielding the QEI.

5. Conceptual and Operational Consequences

QEI provides a bridge between discrete, algebraic invariants (cohomological obstructions in presheaf theory) and global, differential-geometric/topological invariants (index theory). Physically, QEI detects the impossibility of reconstructing a global (separable) state from locally consistent quantum marginals—i.e., the presence of truly quantum correlations that cannot be reconciled by classical gluing. The construction is strictly invariant under local unitaries and captures only the inherently non-classical, non-local quantum structure.

In condensed matter physics, QEI allows the definition of “entanglement-filtered” topological numbers (e.g., an entanglement-projected TKNN invariant). These indices are capable of signaling phase transitions that are invisible to ordinary Berry curvature, responding only to entanglement-driven changes in the Hilbert bundle structure. In high-energy theory, presheaf cohomological obstructions provide the mathematical underpinning for corrections to semiclassical gravitational dynamics induced solely by nontrivial entanglement curvature.

6. Experimental and Theoretical Proposals

Experimentally, QEI can in principle be accessed by reconstructing local amplitude families and measuring holonomies or anomalies under adiabatic parameter cycles (e.g., using interferometric or polarization techniques to access Uhlmann holonomy). The response of certain observables to entanglement-only-induced band transitions serves as a direct experimental diagnostic. In quantum many-body and field-theoretic contexts, QEI may give rise to observable corrections to transport or gravitational response, provided that the appropriate bundle and witness construction can be realized in practice.

From a theoretical perspective, the cohomological–index-theoretic formulation of entanglement opens new directions for the paper of global quantum correlations. Potential extensions include higher categorical generalizations, relations to the classification of topological phases, and the formalization of entanglement-induced anomalies in gauge and gravitational theories.

7. Relation to Related Entanglement Indices and Measures

The presheaf cohomology framework and QEI complement and generalize operational or numerical entanglement indices (such as those based on the PPT criterion, connected correlators, or entropic invariants). Unlike measures depending only on eigenvalues or SDP computations, QEI captures global, topological information lost at the level of local or localizable invariants. In regimes where entanglement measures based on tensor decomposability or partial trace fail to distinguish subtle global effects, QEI provides a robust, integer-quantized diagnostic (Ikeda, 6 Nov 2025). The approach stands in contrast to, but is compatible with, geometric and information-theoretic formulations that arise in quantum information and condensed-matter applications.