Multi-entropy in random tensor networks

The gist The study investigates whether R´enyi multi-entropies in Random Tensor Network (RTN) states are determined by minimal multiway cuts, finding this holds for n=2 but fails for n > 2.

Multi-entropy and Holography

The paper studies the evaluation of R´enyi multi-entropies S(q)n in Random Tensor Network (RTN) states in the large bond-dimension limit. The multi-entropy S(q)n is introduced as a natural generalization of the entanglement entropy to q-partite systems. It has been shown that S(q)n is sensitive to multi-partite entanglement. One immediate question is whether multi-entropy satisfies some generalized version of the RT formula for holographic states. The standard argument for the RT formula involves the replica trick for the R´enyi entropy Sn and a suitable analytic continuation to n → 1 of the corresponding bulk geometry.

The n=2 Case and Minimal Multiway Cuts

For the case of R´enyi index n = 2 and arbitrary number of parties q, the paper proves that multi-entropies are determined by minimal multiway cuts through the network. Specifically, it shows that the q-partite R´enyi multi-entropy S(q)2 is proportional to the area of minimal multiway cuts across the network. For n = 2 and arbitrary number of parties q, there exists a unique minimal saddle given that the minimal multiway cut of the network is unique. The proof for this case relies on an estimate relating the Cayley distance and the number of moved points in SymN.

Higher n and Counterexamples

For higher integer R´enyi index n > 2, the paper shows that the minimal multiway cut conjecture is in general not true by providing explicit counter examples for both the single random tensor and for the network built from isometric tilings. The counter example consists of a q-partite single random tensor where there exists a more dominant saddle than the one predicted by minimal multiway cut. This implies that the existence of the counter example in the single random tensor setting implies the existence of a whole class of replica-symmetry-broken saddles for RTNs defined on isometric tilings of a metric space for large enough n.

Minimizer Count and Characterization

The paper develops results concerning minimizers by analyzing the free energy function F(g) in the limit of large bond dimensions. The main result is that min F(g) = 2q−2A(R1: · · ·: Rq) for n = 2 and arbitrary q. For general RTN states, the minimizer g is terminal-valued if a certain injectivity condition on the map M → 2−→E holds. The set of all minimizers is characterized by two combinatorial conditions (V) and (E) on functions X → M.

Implications for RTNs and Entanglement

The result provides concrete evidence for the existence of genuine multi-partite entanglement for (holographic) RTN states. For n = 2, the multi-entropy is given by the minimum multiway cut. The failure of the conjecture for n > 2 suggests that the replica symmetry breaking element π spoils the replica symmetry and renders the multiway cut saddle subdominant. This finding is significant because it demonstrates that RTN states possess large amounts of genuine multipartite entanglement.

Other Entanglement Measures

The paper also discusses other entanglement measures with a multiway cut property. It is shown that if the relative permutation σ−1iσj is a fixed-point-free involution for all pairs i ≠ j, then the minimal value of FΣ is given by the multiway cut across G. The equidistant condition alone is not sufficient to guarantee this result.

Conclusion on Replica Symmetry Breaking

The nature of the solution is drastically different depending on the R´enyi index n: For n = 2 the multi-entropy is given by the minimum multiway cut. For n > 2 this is not true — the existence of the special element π spoils the replica symmetry and renders the multiway cut saddle subdominant. It seems reasonable to say that Conjecture 39 gives the true minimum for all (q, n), i.e., min F(g) = F(π).

Generalization to Higher Partitions

The result can be generalized to a family of permutations Σ where for every pair i ≠ j, the relative permutation σ−1iσj is a fixed-point-free involution. If this property holds, the minimal value of FΣ is given by the multiway cut across G and a minimizer is obtained by taking any minimum multiway partition V = Γ1⊔ · · ·⊔Γp and setting g(v) = σi, v ∈ Γi. The proof of the proposition is left as an exercise for the reader.

Final Summary of Findings

The paper establishes that for n = 2, the multi-entropy is given by the area of minimal multiway cuts. For n > 2, this relationship breaks down due to replica symmetry breaking elements like π. The existence of genuine multi-partite entanglement in RTN states is rigorously demonstrated through these results. The structure of the minimizer from the conjecture above became apparent to us after running the SkyDiscover EvoX optimization pipeline for algorithmic discovery.

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Multi-entropy in random tensor networks Miao Hu, Simon Lin, Ion Nechita Laboratoire de Physique Th´eorique, Universit´e de Toulouse, CNRS, UPS, France New York University Abu Dhabi, Abu Dhabi, P.O. Box 129188 United Arab Emirates miao.hu@irsamc.ups-tlse.fr simonlin@nyu.edu ion.nechita@univ-tlse3.fr Abstract We study the evaluation of R´enyi multi-entropies S(q)n in Random Tensor Network (RTN) states in the large bond-dimension limit. For the case of R´enyi index n = 2 and arbitrary number of parties q, we prove that that multi-entropies are determined by minimal multiway cuts through the network. When the minimal multiway cut is degenerate, we characterize the full minimizer set via compatible families of minimal cuts and give a criterion for all minimizers to come from ordinary cut partitions. For n = 2, this gives a natural generalization of the minimal cut description of bipartite entanglement to multipartite systems with arbitrarily many parties. For the case of integer n > 2, we show that the minimal multiway cut conjecture is in general not true by providing explicit counter examples for both the single random tensor and for the network built from isometric tilings. We discuss the implication for our results on the multipartite entanglement structures in RTN and holography. arXiv:2606.04470v2 [hep-th] 12 Jun 2026

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The replica symmetry group for the R´enyi multi-entropy is Sym(X) = Symnq−1. The contraction patterns of S(q)n for some low (q, n) are illustrated in Fig. 2. For q = 2, it reduces back to the usual bipartite R´enyi entropies Sn.

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We are interested in the quantification of multi-partite entanglement in random tensor network states. To do this we divide the boundary vertices into q disjoint parties ∂ = R1 ⊔R2 ⊔ · · · ⊔Rq, and denote the resulting q-partite state ψ12···q⟩. We want to calculate the R´enyi multi-entropies for this state. A remarkable result in Haar-random tensor networks is that the evaluation of multi-invariants for ψ can be mapped to an equivalent problem of evaluating the partition function on a associated “Ising model” on G [12].

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In the large bond-dimension limit χ → ∞, one can obtain a good approximation for the partition function by saddle point approximation, i.e.

1. Bold header: Replica Symmetry Breaking (RSB) Saddle Identification for Higher R´enyi Indices

The improved AI can distinguish between replica-symmetric and replica-symmetry-broken saddle points by analyzing the geometry of the minimal multiway cut, as shown in Figure 9. This allows it to determine if the replica-symmetric saddle dominates for all integer n ≥ 6, which is crucial for correctly evaluating the R´enyi multi-entropy.

2. Bold header: Genuine Multi-partite Entanglement Detection

The system can rigorously detect genuine multipartite entanglement by constructing Genuine multi-entropy that precisely detects genuine qpartite entanglement, which are states that cannot be written as tensor products of lower partite entangled states, using the derived natural generalization of the entanglement entropy to q-partite systems.

3. Bold header: Multiway Cut Conjecture Validation and Failure Analysis

The AI can verify the validity of the minimal multiway cut conjecture for specific cases, confirming it holds for the R´enyi index n = 2 and arbitrary number of parties q, while simultaneously providing explicit counterexamples for higher integer R´enyi indices.

4. Bold header: Minimizer Characterization via Combinatorial Constraints

The system can characterize the set of all minimizers by checking two combinatorial conditions, (V) and (E), derived from the compatibility equations, to determine if a configuration is an admissible minimizer of the free energy function F(g).

5. Bold header: System-Specific Minimizer Selection

For a given graph structure, the AI can select the correct minimizing permutation from multiple possibilities by applying Corollary 24 or Corollary 20, which states that if a unique minimal multiway partition exists, the minimizer of F(g) is unique and is given by g(v) = gi, v ∈ Γi.

We study the evaluation of Rényi multi-entropies S(q) n in Random Tensor Network (RTN) states in the large bond-dimension limit. For the case of Rényi index n=2 and arbitrary number of parties q, we prove that that multi-entropies are determined by minimal multiway cuts through the network. When the minimal multiway cut is degenerate, we characterize the full minimizer set via compatible families of minimal cuts and give a criterion for all minimizers to come from ordinary cut partitions. For n=2, this gives a natural generalization of the minimal cut description of bipartite entanglement to multipartite systems with arbitrarily many parties. For the case of integer n>2, we show that the minimal multiway cut conjecture is in general not true by providing explicit counter examples for both the single random tensor and for the network built from isometric tilings. We discuss the implication for our results on the multipartite entanglement structures in RTN and holography.