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arXiv:quant-ph/0310097 · IEEE Transactions on Information Theory

The minimum distance problem for two-way entanglement purification

Andris Ambainis, Daniel Gottesman

Published Jan 25, 2006• 24 citations
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Abstract

Entanglement purification takes a number of noisy EPR pairs |00>+|11> and processes them to produce a smaller number of more reliable pairs. If this is done with only a forward classical side channel, the procedure is equivalent to using a quantum error-correcting code (QECC). We instead investigate entanglement purification protocols with two-way classical side channels (2-EPPs) for finite block sizes. In particular, we consider the analog of the minimum distance problem for QECCs, and show that 2-EPPs can exceed the quantum Hamming bound and the quantum Singleton bound. We also show that 2-EPPs can achieve the rate k/n=1-(t/n)log/sub 2/3-h(t/n)-O(1/n) (asymptotically reaching the quantum Hamming bound), where the EPP produces at least k good pairs out of n total pairs with up to t arbitrary errors, and h(x)=-xlog/sub 2/x-(1-x)log/sub 2/(1-x) is the usual binary entropy. In contrast, the best known lower bound on the rate of QECCs is the quantum Gilbert-Varshamov bound k/n/spl ges/1-(2t/n)log/sub 2/3-h(2t/n). Indeed, in some regimes, the known upper bound on the asymptotic rate of good QECCs is strictly below our lower bound on the achievable rate of 2-EPPs.

DOI
10.1109/tit.2005.862089

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Entanglement purification takes a number of noisy EPR pairs |00>+|11> and processes them to produce a smaller number of more reliable pairs. If this is done with only a forward classical side channel, the procedure is equivalent to using a quantum error-correcting code (QECC). We instead investigate entanglement purification protocols with two-way classical side channels (2-EPPs) for finite block sizes. In particular, we consider the analog of the minimum distance problem for QECCs, and show that 2-EPPs can exceed the quantum Hamming bound and the quantum Singleton bound. We also show that 2-EPPs can achieve the rate k/n=1-(t/n)log/sub 2/3-h(t/n)-O(1/n) (asymptotically reaching the quantum Hamming bound), where the EPP produces at least k good pairs out of n total pairs with up to t arbitrary errors, and h(x)=-xlog/sub 2/x-(1-x)log/sub 2/(1-x) is the usual binary entropy. In contrast, the best known lower bound on the rate of QECCs is the quantum Gilbert-Varshamov bound k/n/spl ges/1-(2t/n)log/sub 2/3-h(2t/n). Indeed, in some regimes, the known upper bound on the asymptotic rate of good QECCs is strictly below our lower bound on the achievable rate of 2-EPPs.
Entanglement purification protocols (EPPs) and quantum error-correcting codes (QECCs) provide two ways of protecting quantum states from interaction with the environment. In an EPP, perfectly entangled pure states are extracted, with some yield D, from a mixed state M shared by two parties; with a QECC, an arbitrary quantum state |\ensuremath{\xi}〉 can be transmitted at some rate Q through a noisy channel \ensuremath{\chi} without degradation. We prove that an EPP involving one-way classical communication and acting on mixed state M^(\ensuremath{\chi}) (obtained by sharing halves of Einstein-Podolsky-Rosen pairs through a channel \ensuremath{\chi}) yields a QECC on \ensuremath{\chi} with rate Q=D, and vice versa. We compare the amount of entanglement E(M) required to prepare a mixed state M by local actions with the amounts D1{\mathit{D}}_{1}D1​(M) and D2{\mathit{D}}_{2}D2​(M) that can be locally distilled from it by EPPs using one- and two-way classical communication, respectively, and give an exact expression for E(M) when M is Bell diagonal. While EPPs require classical communication, QECCs do not, and we prove Q is not increased by adding one-way classical communication. However, both D and Q can be increased by adding two-way communication. We show that certain noisy quantum channels, for example a 50% depolarizing channel, can be used for reliable transmission of quantum states if two-way communication is available, but cannot be used if only one-way communication is available. We exhibit a family of codes based on universal hashing able to achieve an asymptotic Q (or D) of 1-S for simple noise models, where S is the error entropy. We also obtain a specific, simple 5-bit single-error-correcting quantum block code. We prove that iff a QECC results in high fidelity for the case of no error then the QECC can be recast into a form where the encoder is the matrix inverse of the decoder. \textcopyright{} 1996 The American Physical Society.
If the number of initial entanglement pairs shared between two parties (Alice and Bob) is infinite, it is known that one-way entanglement purification protocols (1-EPPs) can perform as well as two-way entanglement purification protocols (2-EPPs) in a noisy quantum channel, whose characteristics are equal to those of a classical channel. However, when EPPs are applied in actual quantum information systems, the number of shared initial entanglement pairs is necessarily finite. In this paper, we consider the application of 1-EPPs and 2-EPPs to finite entanglement pairs that are initially shared between two parties over a phase damping channel. In this case, we demonstrate that 2-EPPs outperform 1-EPPs.
In this article, we present a novel idea for entanglement purification with joint measurements, the joint entanglement purification protocol(joint-EPP). In some quantum communication tasks using entangled pairs, one party holds the whole entangled pairs at the final stage, he or she is able to perform joint measurements on the pairs. In this situation the proposed joint-EPP can improve the entanglement purification by allowing a higher tolerated error rate, faster convergence which uses less steps to achieve a high fidelity, and higher production rate which produces more number of good entangled pairs from a given number of bad entangled pairs. We have established the general correspondence between linear classical codes and the joint-EPP. It has been shown that the joint-EPP can correct errors as long as the error threshod is no larger than 0.5. The joint-EPP works even for fidelity less than 0.5 as long as it is larger than 0.25. We give several concrete examples, definitions and applications of the joint-EPP.
One of the fundamental concepts of quantum information theory is that of entanglement purification; that is, the transformation of a partially entangled state into a smaller-dimensional, more completely entangled state. Of particular interest are protocols for entanglement purification (EPPs) that alternate purely local operations with one- or two-way classical communication. In the present work, we consider a more general, but simpler, class of transformations, called separable superoperators. Since every EPP is a separable superoperator, bounds on separable superoperators apply as well to EPPs; we use this fact to give a new upper bound on the rate of EPPs on Bell-diagonal states, and thus on the capacity of Bell-diagonal channels.
Bipartite entanglement purification is the conversion of copies of weakly entangled pairs shared between two separated parties into a smaller number of strongly entangled shared pairs using only local operations and classical communication. Choosing between different entanglement purification protocols generally involves weighing up a trade-off between the ratio of strongly entangled pairs produced to weakly entangled pairs consumed, which we call the rate of the protocol, and the degree of the entanglement of the strongly entangled pairs, typically measured by the fidelity of those pairs to maximally entangled states. By randomly choosing a protocol according to a probability distribution over a list of protocols for each pair we want to produce, we can achieve rates and fidelities not achieved by any of the original protocols. Here, we show how to choose this distribution to maximize the rate at which we produce qubit pairs with a given fidelity to a Bell state or, equivalently, to maximize the fidelity to a Bell state of the qubit pairs produced at a given rate. We investigate both the asymptotic case, where the number of initial pairs goes to infinity, and the finite-size regime, where protocols are restricted to a finite number of weakly entangled pairs.
We show that a relatively simple reasoning using von Neumann entropy inequalities yields a robust proof of the quantum Singleton bound for quantum error-correcting codes (QECC). For entanglement-assisted quantum error-correcting codes (EAQECC) and catalytic codes (CQECC), a type of generalized quantum Singleton bound [Brunet al., IEEE Trans. Inf. Theory 60(6):3073–3089 (2014)] was believed to hold for many years until recently one of us found a counterexample [MG, Phys. Rev. A 103, 020601 (2021)]. Here, we rectify this state of affairs by proving the correct generalized quantum Singleton bound, extending the above-mentioned proof method for QECC; we also prove information-theoretically tight bounds on the entanglement-communication tradeoff for EAQECC. All of the bounds relate block lengthnnnand code lengthkkkfor given minimum distancedddand we show that they are robust, in the sense that they hold with small perturbations for codes which only correct most of the erasure errors of less thandddletters. In contrast to the classical case, the bounds take on qualitatively different forms depending on whether the minimum distance is smaller or larger than half the block length. We also provide a propagation rule: any pure QECC yields an EAQECC with the same distance and dimension, but of shorter block length.