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arXiv:quant-ph/0404111 · Physical Review A

Interpolation of recurrence and hashing entanglement distillation protocols

K. G. H. Vollbrecht, Frank Verstraete

Published Jun 1, 2005• 55 citations
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Abstract

We construct interesting entanglement distillation protocols by interpolating between the recurrence and hashing protocols. This leads to asymptotic two-way distillation protocols, resulting in an improvement of the distillation rate for all mixed Bell diagonal entangled states, even for the ones with very high fidelity. We also present a method for how entanglement-assisted distillation protocol can be converted into nonentanglement-assisted protocols with the same yield.

DOI
10.1103/physreva.71.062325

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We construct interesting entanglement distillation protocols by interpolating between the recurrence and hashing protocols. This leads to asymptotic two-way distillation protocols, resulting in an improvement of the distillation rate for all mixed Bell diagonal entangled states, even for the ones with very high fidelity. We also present a method for how entanglement-assisted distillation protocol can be converted into nonentanglement-assisted protocols with the same yield.
Bennett et al. proposed a family of protocols for entanglement distillation, namely, hashing, recurrence and breeding protocols. The last one is inferior to the hashing protocol in the asymptotic regime and has been investigated little. In this paper, we propose a framework of converting a stabilizer quantum error-correcting code to a breeding protocol, which is a generalization of the previous conversion methods by Luo-Devetak and Wilde. Then, show an example of a stabilizer that gives a breeding protocol better than hashing protocols, in which the finite number of maximally entangled pairs are distilled from the finite number of partially entangled pairs.
We show how to convert a quantum stabilizer code to a one- or two-way entanglement distillation protocol. The proposed conversion method is a generalization of those of Shor–Preskill and Nielsen–Chuang. The recurrence protocol and the quantum privacy amplification protocol are equivalent to the protocols converted from [[2, 1]] stabilizer codes. We also give an example of a two-way protocol converted from a stabilizer better than the recurrence protocol and the quantum privacy amplification protocol. The distillable entanglement by the class of one-way protocols converted from stabilizer codes for a certain class of states is equal to or greater than the achievable rate of stabilizer codes over the channel corresponding to the distilled state, and they can distill asymptotically more entanglement from a very noisy Werner state than the hashing protocol.
The goal of entanglement distillation is to turn a large number of weakly entangled states into a smaller number of highly entangled ones. Practical entanglement distillation schemes offer a trade-off between the fidelity to the target state and the probability of successful distillation. Exploiting such trade-offs is of interest in the design of quantum repeater protocols. Here, we present a number of methods to assess and optimize entanglement distillation schemes. We start by giving a numerical method to compute upper bounds on the maximum achievable fidelity for a desired probability of success. We show that this method performs well for many known examples by comparing it to well-known distillation protocols. This allows us to show optimality for many well-known distillation protocols for specific states of interest. As an example, we analytically prove optimality of the distillation protocol utilized within the Extreme Photon Loss entanglement generation scheme, even in the asymptotic limit. We proceed to present a numerical method that can improve an existing distillation scheme for a given input state, and we present an example for which this method finds an optimal distillation protocol. An implementation of our numerical methods is available as a Julia package.
Entanglement distillation, the process of converting weakly entangled states into maximally entangled ones using Local Operations and Classical Communication (LOCC), is pivotal for robust entanglement-assisted quantum information processing in error-prone environments. A construction based on stabilizer codes offers an effective method for designing such protocols. By analytically investigating the effective action of stabilizer protocols for systems of prime dimension d , we establish a standard form for the output states of recurrent stabilizer-based distillation. This links the properties of input states, stabilizers, and encodings to the properties of the protocol. Based on those insights, we present a novel two-copy distillation protocol, applicable to all bipartite states in prime dimension, that maximizes the fidelity increase per iteration for Bell-diagonal states. The power of this framework and the protocol is demonstrated through numerical investigations, which provide evidence for superior performance in terms of efficiency and distillability of low-fidelity states compared to other well-established recurrence protocols. By elucidating the interplay between states, errors, and protocols, our contribution advances the systematic development of highly effective distillation protocols, enhancing our understanding of distillability.
We provide generalizations of known two-qubit entanglement distillation protocols for arbitrary Hilbert space dimensions. The protocols, which are analogs of the hashing and breeding procedures, are adapted to bipartite quantum states that are diagonal in a basis of maximally entangled states. We show that the obtained rates are optimal, and thus equal to the distillable entanglement, for a $(d\ensuremath{-}1)$-parameter family of rank deficient states. Methods to improve the rates for other states are discussed. In particular, for isotropic states it is shown that the rate can be improved such that it approaches the relative entropy of entanglement in the limit of large dimensions.
We present an asymptotic bipartite entanglement-distillation protocol that outperforms all existing asymptotic schemes. This protocol is based on the breeding protocol with the incorporation of two-way classical communication. Like breeding, the protocol starts with an infinite number of copies of a Bell-diagonal mixed state. Breeding can be carried out as successive stages of partial information extraction, yielding the same result: one bit of information is gained at the cost (measurement) of one pure Bell state pair (ebit). The basic principle of our protocol is at every stage to replace measurements on ebits by measurements on a finite number of copies, whenever there are two equiprobable outcomes. In that case, the entropy of the global state is reduced by more than one bit. Therefore, every such replacement results in an improvement of the protocol. We explain how our protocol is organized as to have as many replacements as possible. The yield is then calculated for Werner states.