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arXiv:2509.00593 · Physical review. D/Physical review. D.

Genuine multientropy, dihedral invariants, and Lifshitz theory

Clément Berthière, Paul Gaudin

Published Jan 14, 2026• 1 citation
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Abstract

Multi-invariants are local-unitary invariants of state replicas introduced as potential probes of multipartite entanglement and correlations in quantum many-body systems. In this paper, we investigate two multi-invariants for tripartite pure states, namely multientropy and dihedral invariant. We compute the (genuine) multientropy for Lifshitz ground states and obtain its analytical continuation to noninteger values of Rényi index. We show that the genuine multientropy can be expressed in terms of mutual information and logarithmic negativity, a relation that also holds for stabilizer states. For general tripartite pure states, we demonstrate that dihedral invariants are related to Rényi reflected entropies. In particular, we show that the dihedral permutations of replicas are equivalent to the reflected construction, or alternatively to the realignment of density matrices.

DOI
10.1103/vcqd-rkmn

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Multi-invariants are local-unitary invariants of state replicas introduced as potential probes of multipartite entanglement and correlations in quantum many-body systems. In this paper, we investigate two multi-invariants for tripartite pure states, namely multientropy and dihedral invariant. We compute the (genuine) multientropy for Lifshitz ground states and obtain its analytical continuation to noninteger values of Rényi index. We show that the genuine multientropy can be expressed in terms of mutual information and logarithmic negativity, a relation that also holds for stabilizer states. For general tripartite pure states, we demonstrate that dihedral invariants are related to Rényi reflected entropies. In particular, we show that the dihedral permutations of replicas are equivalent to the reflected construction, or alternatively to the realignment of density matrices.
We study genuine multi-entropy as a diagnostic of multipartite entanglement in the toric code, which provides a controlled setting for probing multipartite structures in topologically ordered states. Our main question is whether genuine multi-entropy captures information that is not reducible to conventional lower-party entropic data, such as topological entanglement entropy. We first analyze toric-code ground states that admit a stabilizer-state description, where the relevant quantities can be evaluated exactly. We conjecture that, for stabilizer states and q≥4\mathtt{q}\ge4q≥4, the q\mathtt{q}q-partite genuine multi-entropy at replica index n<qn<\mathtt{q}n<q collapses to a linear combination of multi-entropies involving at most q−2\mathtt{q}-2q−2 parties. We establish this pattern explicitly for q=4\mathtt{q}=4q=4 in the toric code stabilizer sector: for n=2,3n=2,3n=2,3, the genuine multi-entropy is proportional to the tripartite information I3I_3I3​ and, for the Kitaev-Preskill partition, contains no independent genuine four-partite information beyond that captured by the topological entanglement entropy. At n=4n=4n=4, however, this reduction breaks down: the genuine multi-entropy is no longer proportional to I3I_3I3​, but remains a topological invariant of the toric-code stabilizer ground states under Kitaev-Preskill-preserving deformations that do not change whether the numbers of vertex- and plaquette-type junctions coincide. For generic non-stabilizer superpositions within the ground-state manifold and for coherent superpositions of local excitations, the low-nnn reduction also fails. These results show that genuine multi-entropy probes multipartite entanglement structure beyond the tripartite information, and hence beyond the topological entanglement entropy in the Kitaev-Preskill partition, whereas for stabilizer states at low replica index it reduces to lower-partite entropic data.
The multientropy and dihedral measures are a class of tractable measures for multipartite entanglement, which are labeled by the Rényi index (or replica number) n as in the Rényi entanglement entropy. The purpose of this article is to demonstrate that these quantities are useful probes of quantum critical points by examining concrete examples. In particular, we compute the multientropy and dihedral measures in the ( 1 + 1 )-dimensional massless free scalar field theory on a lattice and in the transverse-field Ising model. For n = 2 , we find that the numerical results in both lattice theories quantitatively agree with those from conformal field theoretic calculations. For n = 3 and n = 4 , we provide predictions of these measures for the massless scalar field theory.
It is now well-established that classical statistical mechanics emerges from the entanglement structure of quantum chaotic systems, as quantified by the (subsystem) eigenstate thermalization hypothesis (ETH). While this statement rests on the well-studied bipartite entanglement of the eigenstates of such systems, recent years have shown that many-body states are often characterized by multipartite entanglement, raising the question: what are the universal multipartite entanglement features of chaotic eigenstates? We answer this question in part by studying a tripartite generalization of the ergodic bipartition ansatz through the lens of two tripartite entanglement probes, the Rényi tripartite multi-entropy and the Rényi reflected entropy. We derive leading-order analytical expressions for the ensemble averages of both quantities using a saddle-point analysis. Beyond these measures, we obtain the average of the Rényi-2 reflected density matrix, ρAA∗(2)ρ_{AA^*}^{(2)}ρAA∗(2)​,which controls the Rényi-2 reflected entropy. We show that ρAA∗(2)ρ_{AA^*}^{(2)}ρAA∗(2)​ in a chaotic eigenstate is well-described by this ensemble average which, unlike an ordinary reduced density matrix, takes three qualitatively distinct forms across subsystem-size regimes, which may be viewed as a tripartite generalization of subsystem ETH. We benchmark our analytical predictions against exact diagonalization of the mixed-field Ising model.
We study the tri-partite genuine Rényi multi-entropy of fully symmetric Gaussian states as a function of the squeezing parameter for different total number of bosonic modes under different subsystem configurations. Exact results for a few modes are obtained, and we conjecture an asymptotic formula at the limit of large squeezing parameter as a function of the replica index nnn, while also finding subleading subsystem-size dependent terms for n=3n=3n=3 under a specific tripartition. We analytically show that the genuine n=2n=2n=2 Rényi multi-entropy of any pure bosonic Gaussian states is zero for any number of modes and for any tripartition. Furthermore, to investigate the dependence on subsystem size, we plot the curve of genuine Rényi multi-entropy as a function of the number of modes in the subsystems while keeping the total number of modes fixed. We discuss differences and similarities between this result and studies of the multi-entropy curve for Haar random states. In particular, we identify a subsystem size configuration related with the multi-entropy time in an evaporating black hole for which the behavior of the genuine multi-entropy changes.
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.
The entanglement structure of a bipartite pure state is characterized by its entanglement spectrum, or equivalently, by the family of Rényi-kkk entanglement entropies with integer kkk. 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≥3D\geq 3D≥3 of parties, results in a collection of maps indexed by pairs of DDD-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=3D=3D=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≥3D\geq 3D≥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-NNN expectation values of a number of entanglement spreads are computed in the Haar-random state of local dimension NNN.