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Journal of Modern Optics

Entanglement purification and amplification of three-qubit states using two-outcome weak measurements

Xiang-Ping Liao, Mao-Fa Fang, Jianshu Fang

Published May 19, 2014• 1 citation
Publisher

Abstract

We demonstrate a method for the optimal purification of the asymmetric state, the symmetric state and the state from partially entangled pure states using two-outcome weak measurements. We obtain two different final states by setting different measurement strengths for the same initial state. The total success probability is related to the initial pure state parameters. We also show that the entanglement amplification can be achieved both for mixed state and mixed state by applying this local weak measurements. The approach requires a small cost because a lot of copies and two-body or multibody measurements are not needed.

DOI
10.1080/09500340.2014.918201

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We demonstrate a method for the optimal purification of the asymmetric state, the symmetric state and the state from partially entangled pure states using two-outcome weak measurements. We obtain two different final states by setting different measurement strengths for the same initial state. The total success probability is related to the initial pure state parameters. We also show that the entanglement amplification can be achieved both for mixed state and mixed state by applying this local weak measurements. The approach requires a small cost because a lot of copies and two-body or multibody measurements are not needed.
We propose a measurement-based method to produce a maximally-entangled state from a partially-entangled pure state. Our goal can be thought of as entanglement distillation from a single copy of a partially-entangled state. The present approach involves local two-outcome weak measurements. We show that application of these local weak measurements leads to a probabilistic amplification of entanglement. In addition, we examine how the probability to find the maximally-entangled state is related to the entanglement of the input state. We also study the application of our method to a mixed initial state. We show that the protocol is successful if the separable part of the mixed initial state fulfils certain conditions.
Known entanglement purification protocols for mixed states use collective measurements on several copies of the state in order to increase the entanglement of some of them. We address the question of whether it is possible to purify the entanglement of a state by processing each copy separately. While this is possible for pure states, we show that this is impossible, in general, for mixed states. The importance of this result both conceptually and for experimental realization of purification is discussed. We also give explicit invariants of an entangled state of two qubits under local actions and classical communication.
Entanglement purification is an indispensable ingredient in extended quantum communication networks and usually determines the efficiency and communication rate of quantum communication protocols. Different from all existing entanglement purification protocols (EPPs) where two or more copies of low quality mixed entangled states are selected from the same ensemble, here we describe a general and optimal EPP for arbitrary initial mixed states from different ensembles. We show that the successful operation of EPP may not obtain a higher fidelity mixed state, while the discarded source pair, which is usually regarded as a failure in existing EPPs, may have residual entanglement and can be reused to increase the yield of entanglement purification. We give the criterions of both the successful purification to obtain a higher fidelity mixed state and the existence of residual entanglement. Moreover, we reveal that entanglement purification procedure causes some entanglement loss. Finally, we provide an optimal approach to reduce the entanglement loss. This approach can also be used to increase the yield of entanglement purification. Our EPP may have potential application in long-distance quantum communications.
Abstract Maximally entangled states are crucial for various quantum information processing, yet their quality suffers from entanglement loss due to decoherence. In this paper, a direct purification technique that utilizes weak measurement and requires only two copies of the entangled state is proposed. The method involves transforming the damped entangled state with weak measurements and then applying a direct purification process. This approach yields a maximally entangled state with a certain probability. In the context of Bell state purification, two weak measurement‐based operations designed to convert amplitude‐damped Bell states into the desired state for direct purification method are presented. The first operation is applied prior to the noisy channel, while the second is employed after the noisy channel. Through comprehensive analysis, the performances of these two schemes are evaluated and compared and benchmarked them against a pioneering weak measurement‐based scheme. The findings demonstrate that the proposed methods exhibit superior performance in terms of both fidelity and probability. Additionally, a novel technique for purifying multipartite amplitude‐damped Greenberger–Horne–Zeilinger (GHZ) states is introduced, which existing methods cannot address. By employing only two copies of the entangled states and harnessing the unique advantages of weak measurement operations, the method achieves both resource efficiency and effectiveness.
Purification of mixed states in Quantum Mechanics, by which we mean the transformation into pure states, has been viewed as an {\it Operation} in the sense of Kraus et al and explicit {\it Kraus Operators} \cite{kra1,kra2,kra3} have been constructed for two seperate purification protocols. The first one, initially due to Schrodinger \cite{sch} and subsequently elaborated by Sudarshan et al \cite{sudar}, is based on the {\it preservation of probabilities}. We have constructed a second protocol here based on {\it optimization of fidelities}. Both purification protocols have been implemented on a single qubit in an attempt to improve the fidelity of the purified post measurement state of the qubit with the initial pure state. We have considered both {\it complete} and {\it partial} measurements and have established bounds and inequalities for various fidelities. We show that our purification protocol leads to better state reconstruction, most explicitly so, when partial measurements are made.
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.