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arXiv:quant-ph/0110045 · Physical Review A

Necessary and sufficient condition for distillability with unit fidelity from finite copies of a mixed state: The most efficient purification protocol

Ping‐Xing Chen, Lin-Mei Liang, Cheng-Zu Li, Ming-Qiu Huang

Published Aug 21, 2002• 12 citations
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Abstract

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.

DOI
10.1103/physreva.66.022309

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It is well known that any entangled mixed state in 2\ensuremath⨂22\ensuremath{\bigotimes}22\ensuremath⨂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.
Distillation and purification of entanglement is an important field of quantum information both in theory and in experiment. This paper presents some main conclusions in this field, and discusses mainly the distillation of entanglement from finite copies of a mixed state. By using the conceptions of quasiseparable state and distillable subspace, we show a necessary condition and a sufficient and necessary condition of the distillability from finite copies of a mixed state. We also get the most efficient protocols of entanglement distillation for the case of finite copies. Finally we consider the generalization of these results to the case of infinite copies of a mixed 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.
This paper addresses the question of the distillation of entanglement from a finite number of multipartite mixed states. It is shown that, if one can distill a pure entangled state from n copies of a mixed state ${\ensuremath{\sigma}}_{\mathrm{ABC}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}},$ there must be at least a subspace in the whole Hilbert space of all the copies such that the projection of ${\ensuremath{\sigma}}_{\mathrm{ABC}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}}^{\ensuremath{\bigotimes}n}$ onto the subspace is a pure entangled state. We also show that the purification of entanglement or distillation of entanglement can be carried out by local joint projective measurements with the help of classical communication and local general positive operator valued measurements on a single particle, in principle. Finally, we discuss experimental realizability of the entanglement purification.
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.
Purifying noisy entanglement is a protocol that can increase the entanglement of a mixed state (as a source) at the expense of the entanglement of others (such as an ancilla) by collective measurement. A protocol with which one can get a pure entangled state from a mixed state is defined as purifying mixed states. We address a basic question: can one get a pure entangled state from a mixed state? We give a necessary and sufficient condition of purifying a mixed state by fit local operations and classical communication and show that for a class of source states and ancilla states in arbitrary bipartite systems purifying mixed states is impossible by finite rounds of purifying protocols. For $2\ensuremath{\bigotimes}2$ systems, it is proved that arbitrary states cannot be purified by individual measurement. The possible application and meaning of the conclusion are discussed.
A mixed quantum state $\ensuremath{\rho}$ shared between two parties is said to be distillable if, by means of a protocol involving only local quantum operations and classical communication, the two parties can transform some number of copies of $\ensuremath{\rho}$ into a single shared pair of qubits having high fidelity with the maximally entangled state $|{\ensuremath{\varphi}}^{+}\ensuremath{\rangle}=(|00\ensuremath{\rangle}+|11\ensuremath{\rangle})/\sqrt{2}$. In this Letter it is proved that there exist states that are distillable, but for which an arbitrarily large number of copies is required before any distillation procedure can produce a shared pair of qubits with even a small amount of entanglement. Specifically, for every positive integer nnn there exists a state $\ensuremath{\rho}$ that is distillable, but, given nnn or fewer copies of $\ensuremath{\rho}$, every distillation procedure outputting a single shared pair of qubits outputs those qubits in a separable (i.e., unentangled) state.