Invariant Perfect Tensors

TL;DR

This paper introduces invariant perfect tensors (IPTs), explores their existence, construction, and properties, revealing that while some low-valent IPTs exist uniquely, higher-valent IPTs are absent, yet most large-dimensional invariant tensors are asymptotically perfect.

Contribution

The paper defines invariant perfect tensors, proves existence for low valence, and shows their non-existence at four-valent, while demonstrating asymptotic perfection in high dimensions.

Findings

Bivalent IPTs are unique singlet states.

Trivalent IPTs are uniquely given by Wigner's 3j symbol.

No four-valent IPTs exist for any dimension.

Abstract

Invariant tensors are states in the SU(2) tensor product representation that are invariant under the SU(2) action. They play an important role in the study of loop quantum gravity. On the other hand, perfect tensors are highly entangled many-body quantum states with local density matrices maximally mixed. Recently, the notion of perfect tensors recently has attracted a lot of attention in the fields of quantum information theory, condensed matter theory, and quantum gravity. In this work, we introduce the concept of an invariant perfect tensor (IPT), which is a $n$-valent tensor that is both invariant and perfect. We discuss the existence and construction of IPT. For bivalent tensors, the invariant perfect tensor is the unique singlet state for each local dimension. The trivalent invariant perfect tensor also exists and is uniquely given by Wigner's $3j$ symbol. However, we show that, surprisingly, there does not exist four-valent invariant perfect tensors for any dimension. On the contrary, when the dimension is large, almost all invariant tensors are perfect asymptotically, which is a consequence of the phenomenon of concentration of measure for multipartite quantum states.