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arXiv:quant-ph/9805088 · Physical Review Letters

Entangled Mixed States and Local Purification

Adrian Kent

Published Oct 5, 1998• 95 citations
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Abstract

Linden, Massar, and Popescu have recently given an optimization argument to show that a single two-qubit Werner state, or any other mixture of the maximally entangled Bell states, cannot be purified by local operations and classical communications. We generalize their result and give a simple explanation. In particular, we show that no purification scheme using local operations and classical communications can produce a pure singlet from any mixed state of two spin- $1/2$ particles. More generally, no such scheme can produce a maximally entangled state of any pair of finite-dimensional systems from a generic mixed state. We also show that the Werner states belong to a large class of states whose fidelity cannot be increased by such a scheme.

DOI
10.1103/physrevlett.81.2839

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Linden, Massar, and Popescu have recently given an optimization argument to show that a single two-qubit Werner state, or any other mixture of the maximally entangled Bell states, cannot be purified by local operations and classical communications. We generalize their result and give a simple explanation. In particular, we show that no purification scheme using local operations and classical communications can produce a pure singlet from any mixed state of two spin- 1/21/21/2 particles. More generally, no such scheme can produce a maximally entangled state of any pair of finite-dimensional systems from a generic mixed state. We also show that the Werner states belong to a large class of states whose fidelity cannot be increased by such a 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.
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.
We propose a scheme for purification of a general mixed entangled state. In this scheme, we start from a large number of general mixed entangled states and end up, after local operation and classical communication, with a smaller number of Bell diagonal states with higher entanglement. In particular, the scheme can purify one maximally entangled state from two entangled pairs prepared in a class of mixed entangled state. Furthermore we propose a linear optical implementation of the present scheme with polarization beam splitters and photon detectors.
Multipartite entanglement purification is revisited by using the Local operations and classical communications(LOCCs). We demonstrate our idea by considering the tripartite case, i.e. the purification of tripartite entanglement. We express the general tripartite entangled states in a special representation of total spin operators J1232J_{123}^{2}J1232​ and % J_{12}^{2} of tripartite system with eigenvalues 15/4 and 2 respectively. This basis is a \textit{genuine basis} because it consists of of all the genuine entangled states of tripartite system. Our protocol is a recurrence one, and only two copies of the initial mixed tripartite entangled states are needed in each round. It is shown that if the initial fidelity is larger than a threshold 0.4, the purification process will succeed.The yield of the current protocol is higher than the previous multipartite entanglement purification protocols. As a by-product, we can get a bipartite pure Bell state when the purification protocol fails for W state. Our protocol also shows that there may be some special kind of tripartite entanglement which belongs to neither W-type entanglement nor GHZ-type entanglement.
It is well known that any entangled mixed state in $2\ensuremath{\bigotimes}2$ systems can be purified via infinite copies of the mixed state. But, can one distill a pure maximally entangled state from finite copies of a mixed state in any bipartite system by local operation and classical communication? This is more meaningful in practical application. We give a necessary and sufficient condition for this distillability. This condition requires that there exist distillable subspaces. According to this condition, one can judge easily whether a mixed state is distillable or not. We also analyze some properties of distillable subspaces, and discuss the most efficient purification protocol. Finally, we discuss the distillable enanglement of a two-qubit system for the case of finite copies.
We determine the preshared entanglement required for spatially separated parties, restricted to local operations and classical communication, to attain globally optimal probabilistic two-copy purification of arbitrary bipartite pure states under depolarizing noise. In every local dimension d≥2d\ge 2d≥2, one shared maximally entangled qubit pair suffices: local controlled-SWAP operations exactly reproduce the globally optimal successful transformation. Conversely, any finite-dimensional preshared resource state that attains the same benchmark, even under a positive-partial-transpose relaxation, must have entanglement of formation at least one ebit. For pure resources with one ebit, or for two-qubit resources including mixed states, attaining the benchmark is possible only for states with exactly two nonzero Schmidt weights, both equal to 1/21/21/2, namely those locally unitarily equivalent to a maximally entangled qubit pair. These findings provide a benchmark for evaluating the entanglement demands of noise management in quantum networks and modular quantum computers.