Entanglement cohomology for GHZ and W states

TL;DR

This work advances entanglement cohomology as a robust framework for classifying multipartite entanglement. It provides rigorous closed-form expressions for the cohomology dimensions of generalized GHZ and W states, using a simplicial-lemma approach that links to ordinary homology. Beyond Betti numbers, it introduces two LU invariants—a spectrum of the entanglement Laplacian and a set of intersection numbers—exploiting the geometric structure of entanglement forms to reveal finer information, such as Schmidt data across bipartitions and tripartite entanglement signals. The combination of analytic proofs and numerical experiments strengthens the utility of entanglement cohomology as a diagnostic tool for complex quantum correlations and suggests avenues for deeper connections to geometry and invariant theory in quantum information.

Abstract

Entanglement cohomology assigns a graded cohomology ring to a multipartite pure state, providing homological invariants that are stable under local unitaries and characterize inequivalent patterns of entanglement. In this work we derive exact expressions for the dimensions of these cohomology groups in two canonical entanglement classes, generalized GHZ and W states on an arbitrary number of parties and local Hilbert space dimensions, thus proving conjectures of arXiv:1901.02011. Using the additional structure of the Hodge star and wedge product operations, we propose two new classes of local unitary invariants: the spectrum of the natural Laplacian acting on entanglement $k$-forms, and the intersection numbers obtained from wedge products of representatives for cohomology classes. We present numerical experiments which investigate these invariants in particular states, suggesting that they may provide useful quantities for describing multipartite entanglement.

Figures (2)

• Figure 1: For each value of $\alpha$, we compute the eigenvalues of the entanglement Laplacian $\Delta_1^{(\alpha)}$ associated with the state (\ref{['alpha_states']}) that interpolates between $\ket{\mathrm{GHZ}_3}$ and $\ket{\mathrm{W}_3}$. Independently of $\alpha$, every such spectrum includes three zero eigenvalues and three eigenvalues equal to $3$, which represent the upper and lower bounds of the spectrum; we remove these and focus on the three intermediate eigenvalues $\lambda^{(1)} < \lambda^{(2)} < \lambda^{(3)}$. Each of these three non-trivial eigenvalues occurs with multiplicity two, but we plot only one such curve to represent each pair. Because we sort the eigenvalues in increasing order, the two level crossings in the plot appear as points where two eigenvalue curves meet in a cusp.
• Figure 2: We display the experimental values of the eigenvalues $x^{(\Lambda)}$ and $y^{(\Lambda)}$ of the matrix $M_{ij}^{(\Lambda)}$ given in equation (\ref{['spec_intersection']}). For each value of the Schmidt coefficient $\Lambda$, we prepare the state (\ref{['two_qubit_interpolating']}), find a matrix representation for its entanglement Laplacian acting on entanglement $1$-forms, and then diagonalize to find the six independent $1$-forms which are in its kernel. These are harmonic forms $\alpha_i^{(\Lambda)}$ which give unique representatives for the six cohomology classes. We then compute the intersection matrix (\ref{['intersection_matrix']}) and numerically evaluate its eigenvalues, identifying $x^{(\Lambda)}$ and $y^{(\Lambda)}$ according to their multiplicities. These eigenvalues are plotted along with our proposed closed-form expression (\ref{['yLambda_exact']}) for $y^{(\Lambda)}$, while we perform a fit of $x^{(\Lambda)}$ to a sixth-order polynomial in $\Lambda$ and show the result.

Theorems & Definitions (6)

• Conjecture 1
• Conjecture 2
• Lemma 1
• proof
• proof
• proof