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arXiv:2410.12529 · Physical Review A

Hodge theory for entanglement cohomology

Christian Ferko, Eashan Iyer, Kasra Mossayebi, Gregor Sanfey

Published Mar 21, 2025• 2 citations
arXivHTMLPDF

Abstract

We explore and extend the application of homological algebra to describe quantum entanglement, initiated by Mainiero [arXiv:1901.02011], focusing on the Hodge-theoretic structure of entanglement cohomology in finite-dimensional quantum systems. We construct analogs of the Hodge star operator, inner product, codifferential, and Laplacian for entanglement $k$-forms. We also prove that such $k$-forms obey versions of the Hodge isomorphism theorem and Hodge decomposition, and that they exhibit Hodge duality. As a corollary, we conclude that the dimensions of the $k\mathrm{th}$ and $(n\ensuremath{-}k)\text{\ensuremath{-}}\mathrm{th}$ cohomologies coincide for entanglement in $n$-partite pure states, which explains a symmetry property (``Poincar\'e duality'') of the associated Poincar\'e polynomials.

DOI
10.1103/physreva.111.032422

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We explore and extend the application of homological algebra to describe quantum entanglement, initiated by Mainiero [arXiv:1901.02011], focusing on the Hodge-theoretic structure of entanglement cohomology in finite-dimensional quantum systems. We construct analogs of the Hodge star operator, inner product, codifferential, and Laplacian for entanglement kkk-forms. We also prove that such kkk-forms obey versions of the Hodge isomorphism theorem and Hodge decomposition, and that they exhibit Hodge duality. As a corollary, we conclude that the dimensions of the kthk\mathrm{th}kth and (n\ensuremath−k)\ensuremath-th(n\ensuremath{-}k)\text{\ensuremath{-}}\mathrm{th}(n\ensuremath−k)\ensuremath-th cohomologies coincide for entanglement in nnn-partite pure states, which explains a symmetry property (``Poincar'e duality'') of the associated Poincar'e polynomials.
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 kkk-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.
This talk will explain the structure of Khovanov homology for knots and links and will present a quantum algorithm for its computation that is based on the use of combinatorial Hodge theory. By using combinatorial Hodge theory we can formulate the problem of computing the homology in terms of the determination of the dimension of the eigenvalue one subspace of a unitary operator that is associated with the boundary and coboundary operators of the Khovanov complex. We then use the quantum phase estimation algorithm to find the dimension of this subspace. This is a first foray into the problems of producing a quantum algorithm for the significant knot and link invariant Khovanov homology. There are other deeper problems related to the structure of the chain complex and we shall discuss these difficulties. This paper represents joint work with Sam Lomonaco and Nadya Shirakova.
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This paper is an introduction to work motivated by the question "can multipartite entanglement be detected by homological algebra?" We introduce cochain complexes associated to multipartite density states whose cohomology detects factorizability. The kkkth cohomology components of such cochain complexes produce tuples of (k+1)(k+1)(k+1)-body operators that are non-locally correlated due to the non-factorizability of the state. Associated Poincare polynomials are invariant under local invertible linear transformations (automorphisms that decompose as tensor products). These complexes can be considered as a step toward realizing mutual information as an Euler characteristic. We motivate the definition of the "state index" associated to a multipartite state: a three-parameter function which is: invariant under local invertible transformations, well-behaved under tensor products of states, and interpolates between multipartite mutual information and the integer-valued Euler characteristics of our complexes. We compute cohomologies and state indices of multipartite W and GHZ states. The approach in this paper is directed toward practitioners of finite-dimensional quantum mechanics, although the machinery developed generalizes far beyond. Some results are applicable in infinite dimensions and should be generalizable to the context of quantum field theory. In order to compensate for the long length, a detailed summary is provided in the introduction section.
Provided a maximally entangled state vector in H⊗H{\mathcal{H}}\otimes{\mathcal{H}}H⊗H, the Choi-Jamiolkowski isomorphism of quantum information theory (or channel/state duality) produces a mapping of CP(H,K){\mathsf CP}({\mathcal{H}},{\mathcal{K}})CP(H,K) (completely positive maps of density operators on H{\mathcal{H}}H tothose on K{\mathcal{K}}K) onto the density operators on K⊗H{\mathcal{K}}\otimes{\mathcal{H}}K⊗H, which extends to a unitary map. Here, H{\mathcal{H}}H and K{\mathcal{K}}K are {finite-dimensional} Hilbert spaces.The nature of this construction, the form of the associated formulas, and the possibility of versions which do not require an entangled vector seed, are illuminated from the perspective of {naturality}, in the sense of Eilenberg and MacLane.There is a unique such correspondence (``Choi map'') natural in both H{\mathcal{H}}H and K{\mathcal{K}}K which produces density operators on either L(H,K){\mathsf L}({\mathcal{H}},{\mathcal{K}})L(H,K) (Hilbert-Schmidt inner product) or K⊗H∗{\mathcal{K}}\otimes{\mathcal{H}}^*K⊗H∗ instead of K⊗H{\mathcal{K}}\otimes{\mathcal{H}}K⊗H. In particular, no entangled state input is required. There are no such binatural maps carried on K⊗H{\mathcal{K}}\otimes{\mathcal{H}}K⊗H, but those natural in only K{\mathcal{K}}K are canonically associated with unitary maps H→H∗{\mathcal{H}}\rightarrow{\mathcal{H}}^*H→H∗ and derive from the binatural maps. The construction recovers the standard Choi-Jamiolkowski formulae and shows how they express naturality.