Entanglement distillation using Schur-Weyl decomposition for three qubits

TL;DR

This paper investigates the rates of entanglement distillation in three-qubit systems using Schur-Weyl decomposition, classifying states via SLOCC equivalence, and provides formulas for key coefficients affecting entanglement concentration.

Contribution

It introduces a novel approach to analyze multipartite entanglement rates using Schur-Weyl decomposition and effective Kronecker coefficients, advancing understanding of entanglement classification.

Findings

Asymptotic rates for invariant subspace probabilities are derived.

A combinatorial formula for effective Kronecker coefficients is provided.

The work enhances understanding of entanglement conveyance in multipartite systems.

Abstract

The aim of this work is to examine the exponential rates at which entanglement distillation occur in three-qubits systems. The approach we will follow to elucidate the entanglement concentration is based on the Schur-Weyl decomposition and the Keyl-Werner theorem. In order to clearly state the main results of this work we will have to introduce the notion of SLOCC equivalence and the covariant algebra which will allow us to classify states in different \emph{entanglement classes}. Our main results comprehend the asymptotic rates for the probability of being in an invariant subspace in the Wedderburn decomposition of the state with effective Kronecker coefficient $g_{\alpha\beta\gamma}=1$. We will also present a combinatorial formula for the effective Kronecker coefficient for two-row Young diagrams. With this work we hope to shed some light in the way a multipartite system is entangled and at which rate can we convey entanglement for further applications.