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arXiv:1109.3904 · Journal of Physics A Mathematical and Theoretical

Distillation of entanglement by projection on permutationally invariant subspaces

Mikołaj Czechlewski, Andrzej Grudka, Michał Horodecki, Marek Mozrzymas, Michał Studziński

Published Mar 5, 2012• 4 citations

Abstract

We consider distillation of entanglement from two-qubit states which are mixtures of two pure entangled states and one pure product state, which is orthogonal to them. We distill entanglement from such states by projecting n copies of the state on a permutationally invariant subspace and then applying one-way hashing protocol. We find analytical expressions for the rate of the protocol and show that for wide range of parameters, this protocol achieves higher rates than previous ones. 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. the symmetric group. We have solved this problem in two ways: (i) directly by the use of Schur–Weyl decomposition and Young symmetrizers and (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.

DOI: 10.1088/1751-8113/45/12/125303

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We consider distillation of entanglement from two-qubit states which are mixtures of two pure entangled states and one pure product state, which is orthogonal to them. We distill entanglement from such states by projecting n copies of the state on a permutationally invariant subspace and then applying one-way hashing protocol. We find analytical expressions for the rate of the protocol and show that for wide range of parameters, this protocol achieves higher rates than previous ones. 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. the symmetric group. We have solved this problem in two ways: (i) directly by the use of Schur–Weyl decomposition and Young symmetrizers and (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.

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_{αβγ}=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.

Bipartite qubit entanglement distillation is the process of converting noisy ebits into pure ebits using only local operations and classical communication. This is a core operation for quantum repeaters, enabling such crucial tasks as long-distance quantum communication and distributed quantum computing. In this work, we introduce a method for searching for entanglement distillation protocols and, using this technique, distil qubit Werner states at a higher rate than could be achieved using previously discovered protocols. In particular, we demonstrate the advantage of our new distillation strategy by improving the best-known lower bound for the two-way-assisted quantum capacity of the qubit depolarising channel across a wide range of channel parameters, making progress in one of the long-standing problems of quantum information theory.

We provide generalizations of known two-qubit entanglement distillation protocols for arbitrary Hilbert space dimensions. The protocols, which are analogs of the hashing and breeding procedures, are adapted to bipartite quantum states that are diagonal in a basis of maximally entangled states. We show that the obtained rates are optimal, and thus equal to the distillable entanglement, for a $(d\ensuremath{-}1)$-parameter family of rank deficient states. Methods to improve the rates for other states are discussed. In particular, for isotropic states it is shown that the rate can be improved such that it approaches the relative entropy of entanglement in the limit of large dimensions.

We find that the asymptotic entanglement of assistance of a general bipartite mixed state is equal to the smaller of its two local entropies. Our protocol gives rise to the asymptotically optimal Einstein-Podolsky-Rosen (EPR) pair distillation procedure for a given tripartite pure state, and we show that it actually yields EPR and Greenberger-Horne-Zeilinger (GHZ) states; in fact, under a restricted class of protocols, which we call ``one-way broadcasting,'' the GHZ rate is shown to be optimal. This result implies a capacity theorem for quantum channels where the environment helps transmission by broadcasting the outcome of an optimally chosen measurement. We discuss generalizations to

m

parties and show (for

m=4

) that the maximal amount of entanglement that can be localized between two parties is given by the smallest entropy of a group of parties of which the one party is a member, but not the other. This gives an explicit expression for the asymptotic localizable entanglement and shows that any nontrivial ground state of a spin system can be used as a perfect quantum repeater if many copies are available in parallel. Finally, we provide evidence that any unital channel is asymptotically equivalent to a mixture of unitaries and any general channel to a mixture of partial isometries.