Monogamy of highly symmetric states

TL;DR

This paper explores the limitations of entanglement sharing among multiple particles in highly symmetric quantum states, providing exact bounds and resolving open problems using advanced mathematical tools.

Contribution

It introduces a novel semi-definite programming approach combined with representation theory to determine maximum entanglement projections in symmetric states.

Findings

Exact maximum values for projections onto maximally entangled states

Resolution of long-standing open problems in quantum extendibility

Application of SDP duality and representation theory to quantum information

Abstract

We investigate the extent to which two particles can be maximally entangled when they are also similarly entangled with other particles on a complete graph, focusing on Werner, isotropic, and Brauer states. To address this, we formulate and solve optimization problems that draw on concepts from many-body physics, computational complexity, and quantum cryptography. We approach the problem by formalizing it as a semi-definite program (SDP), which we solve analytically using tools from representation theory. Notably, we determine the exact maximum values for the projection onto the maximally entangled state and the antisymmetric Werner state, thereby resolving long-standing open problems in the field of quantum extendibility. Our results are achieved by leveraging SDP duality, the representation theory of symmetric, unitary and orthogonal groups, and the Brauer algebra.