0
Tree-like pairings guarantee factorization of tensor invariants at large N
Large N factorization of families of tensor trace-invariants
The criterion identifies families whose moments factorize, allowing explicit computation of typical multipartite Renyi entropies in random量子
full image
abstract click to expand
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 R\'enyi entanglement entropy, whose typical expectation value in the uniform random quantum state can be explicitly computed assuming large N factorization.
