Random Invariant Tensors

TL;DR

This paper studies the entanglement properties of random invariant tensors in SU(2) representations, revealing that their entropy deficits remain finite for higher valence tensors, contrasting with non-invariant cases.

Contribution

It demonstrates that the expected entropy deficit of n-valent random invariant tensors remains finite for n ≥ 5, unlike the divergent deficit in non-invariant cases.

Findings

Expected entropy deficit is finite for n ≥ 5

Invariant tensors exhibit concentration of measure in entanglement entropy

Entropy deficit can be smaller than half a bit under certain conditions

Abstract

Invariant tensors are states in the (local) SU(2) tensor product representation but invariant under global SU(2) action. They are of importance in the study of loop quantum gravity. A random tensor is an ensemble of tensor states. An average over the ensemble is carried out when computing any physical quantities. The random tensor exhibits a phenomenon of `concentration of measure', saying that for any bipartition, the expected value of entanglement entropy of its reduced density matrix is asymptotically the maximal possible as the local dimension goes to infinity. This is also true even when the average is over the invariant subspace instead of the whole space for $4-$valent tensors, although its entropy deficit is divergent. One might expect that for $n\geq 5$, $n-$valent random invariant tensor would behavior similarly. However, we show that, the expected entropy deficit of reduced density matrix of such $n-$valent random invariant tensor from maximum, is not divergent but a finite number. Under some special situation, the number could be even smaller than half a bit, which is the deficit of random pure state over the whole Hilbert space from maximum.