Distillation of entanglement by projection on permutationally invariant subspaces

TL;DR

This paper presents a method for entanglement distillation from specific two-qubit mixed states using projection on permutationally invariant subspaces, with analytical rate expressions and generalizations to higher dimensions.

Contribution

It introduces a novel entanglement distillation technique based on permutational invariance and provides analytical formulas for the protocol's rate, extending to higher-dimensional systems.

Findings

Analytical expressions for distillation rate in two-qubit systems.

Method successfully generalizes to higher-dimensional quantum systems.

Mathematical solutions for diagonalizing symmetry-invariant matrices.

Abstract

We consider distillation of entanglement from two qubit states which are mixtures of three mutually orthogonal states: two pure entangled states and one pure product state. We distill entanglement from such states by projecting n copies of the state on permutationally invariant subspace and then applying one-way hashing protocol. We find analytical expressions for the rate of the protocol. We also generalize this method to higher dimensional systems. To get analytical expression for two qubit case, we faced a mathematical problem of diagonalizing a family of matrices enjoying some symmetries w.r.t. to symmetric group. We have solved this problem in two ways: (i) directly, by use of Schur-Weyl decomposition and Young symmetrizers (ii) showing that the problem is equivalent to a problem of diagonalizing adjacency matrices in a particular instance of a so called algebraic association scheme.