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arXiv:0811.4046 · Physical Review A

Entanglement purification protocol for a mixture of a pure entangled state and a pure product state

Mikołaj Czechlewski, Andrzej Grudka, Satoshi Ishizaka, Antoni Wójcik

Published Jul 30, 2009• 14 citations
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Abstract

We present an entanglement purification protocol for a mixture of a pure entangled state and a pure product state, which are orthogonal to each other. The protocol is a combination of bisection method and one-way hashing protocol. We give recursive formula for the rate of the protocol for different states, i.e., the number of maximally entangled two-qubit pairs obtained with the protocol per a single copy of the initial state. We also calculate numerically the rate for some states.

DOI
10.1103/physreva.80.014303

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We present an entanglement purification protocol for a mixture of a pure entangled state and a pure product state, which are orthogonal to each other. The protocol is a combination of bisection method and one-way hashing protocol. We give recursive formula for the rate of the protocol for different states, i.e., the number of maximally entangled two-qubit pairs obtained with the protocol per a single copy of the initial state. We also calculate numerically the rate for some states.
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.
We find the necessary and sufficient condition under which two two-qubit mixed states can be purified into a pure maximally entangled state by local operations and classical communication. The optimal protocol for such transformation is obtained. This result leads to a necessary and sufficient condition for the exact purification of nnn copies of a two-qubit state.
Bipartite entanglement purification is the conversion of copies of weakly entangled pairs shared between two separated parties into a smaller number of strongly entangled shared pairs using only local operations and classical communication. Choosing between different entanglement purification protocols generally involves weighing up a trade-off between the ratio of strongly entangled pairs produced to weakly entangled pairs consumed, which we call the rate of the protocol, and the degree of the entanglement of the strongly entangled pairs, typically measured by the fidelity of those pairs to maximally entangled states. By randomly choosing a protocol according to a probability distribution over a list of protocols for each pair we want to produce, we can achieve rates and fidelities not achieved by any of the original protocols. Here, we show how to choose this distribution to maximize the rate at which we produce qubit pairs with a given fidelity to a Bell state or, equivalently, to maximize the fidelity to a Bell state of the qubit pairs produced at a given rate. We investigate both the asymptotic case, where the number of initial pairs goes to infinity, and the finite-size regime, where protocols are restricted to a finite number of weakly entangled pairs.
We find the necessary and sufficient condition under which two two-qubit mixed states can be purified into a pure maximally entangled state by local operations and classical communication. The optimal protocol for such transformation is obtained. This result leads to a necessary and sufficient condition for the exact purification of nnn copies of a two-qubit state.
Entanglement purification allows the creation of qubit pairs of arbitrarily high fidelity with respect to a maximally entangled state, starting from a larger number of low-fidelity pairs. Purification requires quantum memory, a role for which electron spins are well suited. However, using existing recurrence protocols involving symmetric local two-qubit operations for spin qubits turns out to be rather unpractical. We present an efficient purification protocol requiring only a single pulsed Heisenberg- or XY-type exchange interaction between two qubit pairs. In contrast to known protocols, we allow for asymmetric bilateral two-qubit operations where the two communication parties operate differently on their qubits. In the optimal version of our protocol, the local two-qubit interactions in the case of Heisenberg exchange correspond to the swap\sqrt{swap}swap​ gate and its inverse.