Extendibility of Werner States

TL;DR

This paper provides a comprehensive analytical characterization of the two-sided symmetric extendibility of Werner states, revealing complex trade-offs and conditions based on symmetry and representation theory.

Contribution

It introduces a novel analytical approach to determine extendibility of Werner states using SU(d) representation theory, solving a previously complex problem.

Findings

Necessary and sufficient conditions for Werner state extendibility.

Trade-off between extension sizes on two sides.

Comparison with isotropic states extendibility.

Abstract

We investigate the two-sided symmetric extendibility problem of Werner states. The interplay of the unitary symmetry of these states and the inherent bipartite permutation symmetry of the extendibility scenario allows us to map this problem into the ground state problem of a highly symmetric spin-model Hamiltonian. We solve this ground state problem analytically by utilizing the representation theory of SU(d), in particular a result related to the dominance order of Young diagrams in Littlewood-Richarson decompositions. As a result, we obtain necessary and sufficient conditions for the extendibility of Werner states for arbitrary extension size and local dimension. Interestingly, the range of extendible states has a non-trivial trade-off between the extension sizes on the two sides. We compare our result with the two-sided extendibility problem of isotropic states, where there is no such trade-off.