Extendibility of Brauer states

TL;DR

This work develops a representation-theoretic framework for the extendibility of Brauer states, integrating Schur-Weyl duality for $ ext{U}(d)$ and $ ext{O}(d)$ and Brauer algebra techniques to characterize $(n,m)$-extendible and $n$-de Finetti-extendible Brauer states. It provides a general recipe based on commutants and averaged observables $f_{n,m}$ and $b_{n,m}$, expressed via unitary and orthogonal Casimir operators, to identify the convex parameter sets in the $( ext{Tr}( ho F), ext{Tr}( ho B))$ plane. The paper delivers explicit results for $(1,2)$, $(1,3)$ and $(2,2)$ extendibility in all dimensions, an estimate for $(1,m)$-extendibility, and a complete analysis of $n$-de Finetti-extendibility, including exact finite-$n$ and a detailed $n oty$ limiting shape that is not a polygon for odd dimensions. These findings illuminate how Brauer-state extendibility scales with dimension and extendibility number, and they align with and extend previous results on Werner and isotropic states. The methods offer a principled pathway to study higher-order extendibility and related symmetry-protected state families across quantum information and many-body physics.

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.

Figures (20)

• Figure 1: The set of Brauer states at $d=3$ in the $\mathop{\mathrm{Tr}}\nolimits(\rho\mathbf{F}) - \mathop{\mathrm{Tr}}\nolimits(\rho\mathbf{B})$ parametrisation appearing as the grey triangle. The set of Werner states is illustrated as the dashed line, while the set of isotropic states as the dotted line. The set of separable Brauer states constitute the grey rectangle.
• Figure 2: Illustration of the two types of diagonalisation processes of the represented Casimir operators, starting from the same point. On the left side the irreps are fused together and then a restriction happens to the subalgebra $\mathfrak{so}(d)$, while on the right side the process is done the other way around.
• Figure 3: The set of Brauer states at $d=2,3,4$ and $d\to\infty$ in the $\mathop{\mathrm{Tr}}\nolimits(\rho\mathbf{F}) - \mathop{\mathrm{Tr}}\nolimits(\rho\mathbf{B})$ parametrisation appearing as the grey triangle. In most figures subset of $(1,2)$- and $(2,2)$-extendible Brauer states appears in red in the background, and the subset of $(1,3)$-extendible Brauer states appears in yellow in the foreground. However, in the $d=3$ case the subset of $(2,2)$-extendible Brauer states appears in blue as it does not coincide with the subset of $(1,2)$-extendible Brauer states. The set of Werner states is illustrated as the dashed line, while the set of isotropic states as the dotted line. The set of separable Brauer states constitute the grey rectangle. However, in the $d\to\infty$ case the set of Werner states and the set of separable Brauer states have merged into the lower side.
• Figure 4: The set of Brauer states at $d=2,3,4$ and $5$ in the $\mathop{\mathrm{Tr}}\nolimits(\rho\mathbf{F}) - \mathop{\mathrm{Tr}}\nolimits(\rho\mathbf{B})$ parametrisation appearing as the grey triangle. The subset of $(1,2)$-extendible Brauer states appears in red in the background, and the subset of $(1,3)$-extendible Brauer states appears in yellow in its foreground. The estimates for the sets of $(1,2)$- and $(1,3)$-extendible Brauer states, appear as the blue polygons with differing shades, contained in each other. The set of Werner states is illustrated as the dashed line, while the set of isotropic states as the dotted line. The set of separable Brauer states constitute the grey rectangle.
• Figure 5: The set of Brauer states at $d=3,10$ and $d\to\infty$ in the $\mathop{\mathrm{Tr}}\nolimits(\rho\mathbf{F}) - \mathop{\mathrm{Tr}}\nolimits(\rho\mathbf{B})$ parametrisation appearing as the grey triangle. The subset of $(1,2)$-extendible Brauer states appears in red in the background, and the subset of $(1,3)$-extendible Brauer states appears in yellow in its foreground. The estimates of the sets of $(1,m)$-extendible Brauer states, $m\in[2,10]$, appear as the blue polygons with differing shades, contained in each other. The set of Werner states is illustrated as the dashed line, while the set of isotropic states as the dotted line. The set of separable Brauer states constitute the grey rectangle. However, in the $d\to\infty$ case the set of Werner states and the set of separable Brauer states have merged into the lower side.

Theorems & Definitions (53)

• Definition 1
• Theorem 1
• Theorem 2
• Definition 2
• Definition 3
• Lemma 1
• proof
• Theorem 3
• Theorem 4
• Theorem 5