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arXiv:quant-ph/0510189 · Chinese Physics Letters

Entanglement Distillation for Mixed States Using Particle Statistics

Huang Xiao-Li, Cheng Li-hong, YI Xue-Xi

Published Mar 30, 2006• 3 citations
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Abstract

We extend the idea of entanglement concentration protocol for pure states (Phys. Rev. Lett. 88 , 187903) to the case of entanglement distillation for mixed states. The scheme works only with particle statistics and local operations, without the need of any other interactions. We show that the maximally entangled state can be distilled out when the initial state is pure, otherwise the entanglement distilled depends on the off-diagonal element of density matrix of the initial state. Because of the requirement for density matrix, the entanglement distilled is always less than one, and this result is the same for both fermionic and bosonic particles. The distillation efficiency is a product of the diagonal elements of the initial state, it takes the maximum 50%, the same as that in the case for pure states.

DOI
10.1088/0256-307x/23/4/003

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We extend the idea of entanglement concentration protocol for pure states (Phys. Rev. Lett. 88 , 187903) to the case of entanglement distillation for mixed states. The scheme works only with particle statistics and local operations, without the need of any other interactions. We show that the maximally entangled state can be distilled out when the initial state is pure, otherwise the entanglement distilled depends on the off-diagonal element of density matrix of the initial state. Because of the requirement for density matrix, the entanglement distilled is always less than one, and this result is the same for both fermionic and bosonic particles. The distillation efficiency is a product of the diagonal elements of the initial state, it takes the maximum 50%, the same as that in the case for pure states.
Abstract Production of quantum states exhibiting a high degree of entanglement out of noisy conditions is one of the main goals of quantum information science. Here, we provide a conditional yet efficient entanglement distillation method which functions within the framework of spatially localized operations and classical communication. This method exploits indistinguishability effects due to the spatial overlap between two identical qubits in distinct sites and encompasses particle statistics imprint. We derive the general conditions for the maximum entanglement distillation out of mixed states. As applications, we give a thorough description of distilled entanglement and associated success probability starting from typical noisy states, such as thermal Gibbs states and Werner states. The influence of local temperatures and of a noise parameter is discussed, respectively, in these two cases. The proposed scheme paves the way towards quantum repeaters in composite networks made of controllable identical quantum particles.
In this paper, we propose two general entanglement distillation protocols which can concentrate the state (non-maximally entangled pure W state). The general protocols are mainly based on the unitary transformation on the auxiliary particle and one of the three entangled particles, and in the second protocol, the entanglement distillation includes two meanings, namely, extracting the concentrated tripartite entangled W state and obtaining the maximally entangled bipartite state from the 'garbage' state, which gives no contribution to the distillation of the state. We can make use of the 'garbage' in the distillation process, and make the entanglement waste in the quantum communication as small as possible. A feasible physical scheme is suggested based on the cavity QED.
我们为纯状态扩大纠纷集中协议的想法(Phys。加快。Lett。88， 187903 ) 到为混合状态的纠纷蒸馏的案例。计划仅仅与粒子统计和本地操作工作，没有任何另外的相互作用的需要。当起始的状态是纯的时，我们证明纠缠的格言联盟者声明罐头外面被提取，否则提取的纠纷取决于起始的状态的密度矩阵的离开斜的元素。因为为密度矩阵的要求，提取的纠纷总是是不到一个，并且这结果是为 fermionic 和 bosonic 粒子的一样。Thedistillation 效率是起始的状态的斜元素的一个产品，它拿最大值 50% ，与那一样在为纯状态的情况中。
The information-theoretic approach to entanglement typically relies on arbitrarily many input copies and unboundedly complex operations; however, the available resources are constrained in reality. This motivates the study of entanglement theory with limited resources. One of its central questions is which quantities characterize entanglement manipulations, and so far, only the idealized case of pure states has been well explored. In this work, we address entanglement distillation from mixed states and identify the Rényi coherent information as the quantity determining the optimal rate in the worst case under two distinct resource constraints. We first characterize the sample-efficient regime, in which unbounded computational power is allowed. Our universal protocol achieves this rate without knowledge of the input state under certain polynomial rank constraints. This rate is optimal among universal protocols: we construct states whose distillable entanglement is nearly extensive, yet from which no universal sample-efficient protocol attains a rate above their Rényi coherent information. We then push this characterization to the computationally efficient frontier: the same bound holds for any efficient protocol even when the input state is known, and another universal protocol achieves this rate efficiently under additional polynomial constraints on the marginal ranks. Overall, our results give the Rényi coherent information a novel operational meaning in two computational settings: universal distillation with limited copies and efficient distillation even with knowledge of the input state.
Entanglement distillation is a basic task in quantum information, and the distillable entanglement of three bipartite reduced density matrices from a tripartite pure state has been studied in [Phys. Rev. A 84, 012325 (2011)]. We extend this result to tripartite mixed states by studying a conjectured matrix inequality, namely rank(∑iRi⊗Si)≤Krank(∑iRiT⊗Si)\mathop{\rm rank}(\sum_i R_i \otimes S_i)\leq K \mathop{\rm rank}(\sum_i R_i^T \otimes S_i)rank(∑i​Ri​⊗Si​)≤Krank(∑i​RiT​⊗Si​) holds for any bipartite matrix M=∑iRi⊗SiM=\sum_i R_i \otimes S_iM=∑i​Ri​⊗Si​ and Schmidt rank KKK. We prove that the conjecture holds for MMM with K=3K=3K=3 and some special MMM with arbitrary KKK.
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.