Extendibility of Brauer states

TL;DR

This paper analyzes the extendibility properties of Brauer states using representation theory, providing explicit criteria and parameter sets for various types of extendibility in different dimensions.

Contribution

It offers a general recipe for determining extendibility of Brauer states and explicitly characterizes the parameter sets for specific cases, including asymptotic behavior.

Findings

Explicit parameter sets for (1,2)-, (1,3)-, and (2,2)-extendibility in any dimension.

Trade-offs in extendibility parameters n and m.

Asymptotic shape of the set of n-de Finetti-extendible states for large n.

Abstract

We investigate the extendibility problem for Brauer states, focusing on the symmetric two-sided extendibility and the de Finetti extendibility. By employing the representation theory of the unitary and orthogonal groups, we provide a general recipe for determining the set of $(n,m)$-extendible and $n$-de Finetti-extendible Brauer states. From the concrete form of the commutant of the diagonal action of the orthogonal group, we explicitly determine the set of parameters for which the Brauer states are $(1,2)$-, $(1,3)$- and $(2,2)$-extendible in any dimension $d$ and find that Brauer states extend with a non-trivial trade-off in $n$ and $m$. Using the same recipe we also provide an estimate of the set of $(1,m)$-extendible Brauer states for any $m$ and dimension $d$. Finally, using the branching rules from $\mathrm{U}(d)$ to $\mathrm{O}(d)$, we obtain the set of $n$-de Finetti-extendible Brauer states in any dimension, and also analytically describe the $n\to\infty$ limiting shape which turns out not to be a polygon for odd dimensions.