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Europhysics Letters (EPL)

Low-order maximally single-trace graphs as the first counterexamples to large-N factorization in random tensors

Jonathan Bethold, Hannes Keppler

Published Jul 27, 2026• 0 citations
Publisher

Abstract

Abstract We give the first- and lowest-order examples of 3-regular 3-edge-colored graphs that demonstrate the non-factorization of tensor model invariants in the large- N limit of Gaussian random tensors. The existence of such graphs has recently been proven on general grounds, but without providing explicit examples. Moreover, it was expected that such examples require graphs with a very large number of vertices. We show that the smallest such graphs have 16 vertices. The non-factorization for Gaussian random tensors is in stark contrast to the well-known large- N factorization for random matrices.

DOI
10.1209/0295-5075/ae90e0

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Abstract We give the first- and lowest-order examples of 3-regular 3-edge-colored graphs that demonstrate the non-factorization of tensor model invariants in the large- N limit of Gaussian random tensors. The existence of such graphs has recently been proven on general grounds, but without providing explicit examples. Moreover, it was expected that such examples require graphs with a very large number of vertices. We show that the smallest such graphs have 16 vertices. The non-factorization for Gaussian random tensors is in stark contrast to the well-known large- N factorization for random matrices.
It was recently proven that, in contrast to their matrix analogues, the moments of a real Gaussian tensor of size N do not in general factorize over their connected components in the asymptotic large N limit. While the original proof of this rather surprising result was not constructive, explicit examples of non-factorizing moments, which are expectation values of trace-invariants, have since then been discovered. We explore further aspects of this problem, with a focus on Haar-distributed (or Gaussian) complex random tensors, which are more directly relevant to quantum information. We start out by exhibiting an explicit example of non-factorizing trace-invariant, thereby filling a gap in the recent literature. We then turn to the opposite question: that of finding interesting families of trace-invariants that do in fact factorize at large N. We establish three main theorems in this regard. The first one provides a sufficient combinatorial bound ensuring large N factorization, that is also simple enough to be applicable to various cases of practical relevance. Our second main result shows that the expectation value of any compatible trace-invariant is dominated by certain tree-like combinatorial structures at large N, which we refer to as tree-like dominant pairings. Our third main theorem establishes that any trace-invariant admitting tree-like dominant pairings does actually factorize at large N. In this way, we are able to prove that various families of trace-invariants that have been previously studied in the literature do factorize at large N. We apply our findings to the theory of multipartite quantum entanglement: to any trace-invariant is associated a multipartite generalization of Rényi entanglement entropy, whose typical expectation value in the uniform random quantum state can be explicitly computed assuming large N factorization.
Two direct connections between random tensors and random matrices are discussed in this article. In the rst part, we introduce U( \tau ) matrix models which generate fully packed, oriented loops on random surfaces. e latter are found to be in bijection with a set of regular edge-colored graphs. It is shown that the expansion in the number of loops is organized like the 1/ N expansion of rank-three tensor models. Recent results on tensor models are applied in this context. For example, congurations which maximize the number of loops are precisely the melonic graphs of tensor models and a scaling limit which projects onto themelonic sector is found. is approach is generalized to higher-rank tensor models, which generate loops with fugacity \tau on triangulations in dimension d–1 . In the second part, we introduce singular value decompositions to evaluate the expectations of polynomial observables of Gaussian random tensors. Performing the integrals over the unitary group leads to a notion of eective observables which expand onto regular trace invariants. We show that both asymptotic and exact new calculations of expectations can be performed this way.
We consider the universality for the trace invariants of c1N×⋯×cDNc_1 N \times \cdots \times c_D Nc1​N×⋯×cD​N tensors with i.i.d. complex random elements. In the case c1=⋯=cDc_1 = \cdots = c_Dc1​=⋯=cD​, Gurau derived the universality in the limit N→∞N \to \inftyN→∞ by representing the average trace invariants in terms of the corresponding colored graphs. Moreover he could explicitly calculate the asymptotic forms of the average trace invariants, if the corresponding graphs were special ones called "melonic graphs". In this paper we study another kind of special graphs, cycle graphs with multiple edges. One can construct these graphs by using simple cycle graphs and melonic graphs. Counting the number of the components, we can explicitly calculate the asymptotic forms of the average trace invariants.
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.
The entanglement entropy of a random tensor network (RTN) is studied using tools from free probability theory. Random tensor networks are simple toy models that help in understanding the entanglement behavior of a boundary region in the anti-de Sitter/conformal field theory (AdS/CFT) context. These can be regarded as specific probabilistic models for tensors with particular geometry dictated by a graph (or network) structure. First, we introduce a model of RTN obtained by contracting maximally entangled states (corresponding to the edges of the graph) on the tensor product of Gaussian tensors (corresponding to the vertices of the graph). The entanglement spectrum of the resulting random state is analyzed along a given bipartition of the local Hilbert spaces. The limiting eigenvalue distribution of the reduced density operator of the RTN state is provided in the limit of large local dimension. This limiting value is described through a maximum flow optimization problem in a new graph corresponding to the geometry of the RTN and the given bipartition. In the case of series-parallel graphs, an explicit formula for the limiting eigenvalue distribution is provided using classical and free multiplicative convolutions. The physical implications of these results are discussed, allowing the analysis to move beyond the semiclassical regime without any cut assumption, specifically in terms of finite corrections to the average entanglement entropy of the RTN.
The SYK model involves N Majorana fermions in 1+0 dimensions with quenched Gaussian dis- order and proves to be exactly solvable in the large N limit at strong coupling. It has been initially proposed by Sachdev and Ye as a model of condensed matter and later gained some interest as a toy model of AdS/CFT correspondence, thanks to the work of Kitaev. On the other side, random tensors are generalisations of random matrices to objets that carry more than two indices. It turns out that the SYK model and random tensors involve a special class of Feynman graphs known as "melons". We will briefly review both constructions. Then, we will show how non Gaussian disorder can be reduced to a Gaussian one, treating the coupling as a random tensor, thanks to Gurau’s Gaussian universality result.