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
kth 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.
Organising the space of entanglement structures of a multipartite quantum system is a much more challenging task than its bipartite version: while the local unitary (LU) orbit of a bipartite pure state can be conveniently characterized by its entanglement spectrum, invariants of multipartite entanglement structures are comparatively difficult to define and work with. The root cause of this difference is that the bipartite problem can be reduced to the analysis of matrix invariants, while its multipartite version is governed by a much richer space of tensor invariants. The present work explores the latter through the lens of so-called trace-invariants, which are in one-to-one correspondence with combinatorial objects known as colored graphs. We first explain why trace-invariant evaluations can serve as labels of LU-orbits of multipartite pure states, how this strategy extends to random states, and how the effect of local operations (LO) can be analyzed through such data. We then focus on entanglement classification within an (infinite-dimensional) subspace of reference states, whose basic building blocks are GHZ states of various dimensions. We show that relatively simple subclasses of trace-invariants are sufficient to separate the LU-orbits of reference states, and enable a complete (resp. an incomplete) characterization of their relations in the LO (resp. LOCC) resource theory of entanglement. Finally, we investigate how a (still infinite) subclass of reference states of local dimension N can be efficiently distinguished at leading and subleading orders in an asymptotic large-N expansion (among themselves, or from Haar-random states). This analysis relies crucially on combinatorial quantities associated to colored graphs, some of which have already played instrumental roles in the recent literature on random tensors. Results of broader relevance are reported along the way.
Local unitary equivalence is an important ingredient for quantifying and classifying entanglement. Verifying whether or not two quantum states are local unitary equivalent is a crucial problem, where only the case of multipartite pure states is solved. For mixed states, however, the verification of local unitary equivalence is still a challenging problem. In this paper, based on the coefficient matrices of generalized Bloch representations of quantum states, we find a variety of local unitary invariants for arbitrary-dimensional bipartite quantum states. These invariants are operational and can be used as necessary conditions for verifying the local unitary equivalence of two quantum states. Furthermore, we extend the construction to the arbitrary-dimensional multipartite case. We finally apply these invariants to estimate concurrence, a vital entanglement measure, showing the practicability of local unitary invariants in characterizing entanglement.
The entanglement structure of a bipartite pure state is characterized by its entanglement spectrum, or equivalently, by the family of Rényi-
k entanglement entropies with integer
k. Taking the difference of two such entropies defines a quantity known as an entanglement spread, that is monotonous under local operations, and that vanishes if and only if the entanglement spectrum of the state is flat. We introduce a multipartite generalization of this notion which, for a fixed number
D≥3 of parties, results in a collection of maps indexed by pairs of
D-edge-colored graphs (obeying some condition). Each member of this collection takes the form of a difference of two multipartite Rényi entanglement entropies, each associated to a local unitary polynomial invariant (known in this context as a trace-invariant), and is shown to be monotonous under local operations. We then argue that a previously introduced family of so-called hypergraph-tensor (HT) states can be understood as a multipartite counterpart to the family of flat bipartite states. Indeed, we first prove that, when
D=3, a tripartite pure state is HT if and only if it minimizes a particular (infinite) family of entanglement spreads. Second, we embed the set of HT states into the much larger family of spectral hypergraph-tensor (SHT) states, defined by a new Ansatz we introduce. We then prove that HT states can be uniquely characterized as the minimizers of some fixed entanglement spread among SHT states, for arbitrary
D≥3. We also investigate SHT states in their own right: in particular, we characterize their orbits under local unitary transformations. Finally, along the way, the asymptotic large-
N expectation values of a number of entanglement spreads are computed in the Haar-random state of local dimension
N.
Multipartite entanglement is a natural generalization of bipartite entanglement, but is relatively poorly understood. In this paper, we develop tools to calculate a class of multipartite entanglement measures - known as multi-invariants - for stabilizer states. We give an efficient numerical algorithm that computes multi-invariants for stabilizer states. For tripartite stabilizer states, we also obtain an explicit formula for any multi-invariant using the GHZ-extraction theorem. We then present a counting argument that calculates any Coxeter multi-invariant of a q-partite stabilizer state. We conjecture a closed form expression for the same. We uncover hints of an interesting connection between multi-invariants, stabilizer states and topology. We show how our formulas are further simplified for a restricted class of stabilizer states that appear as ground states of interesting models like the toric code and the X-cube model.
We develop a cohomological framework for entanglement entropy that unifies information theory, operator algebras, and noncommutative geometry. Entropy is a 1-cocycle in information cohomology; we generalize this to the quantum setting via Hochschild and cyclic cohomology. For an inclusion of von Neumann algebras N contained in M with a faithful normal state w admitting a cyclic and separating vector, and a conditional expectation E: M -> N (whose existence is equivalent to modular invariance of N by Takesaki's theorem), we construct the entanglement complex as the image of the transgression map tau_w on the relative cyclic complex, where A is a dense *-subalgebra of M and B = A intersect N. Our main result identifies entanglement cohomology with the kernel of the restriction map HC*(A) -> HC*(B), up to the connecting homomorphism d of the long exact sequence of the pair (A,B); when d = 0, the identification is exact. The relative modular cocycle u_t = (Dw : D(w|_N) o E)_t generates explicit entanglement classes when E is not w-invariant, with the Araki relative entropy as its infinitesimal generator. The framework accommodates Type III von Neumann algebras, where no local density matrix exists, and leads to a holographic proposal: a subspace of H^1_ent is conjecturally dual to linearized modular Hamiltonian perturbations of the bulk entanglement wedge. We verify the construction in finite-dimensional, type II_1, and type III_1 examples.