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arXiv:quant-ph/9707002

Entanglement purification via separable superoperators

Eric M. Rains

Published Jul 1, 1997• 47 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.quant-ph/9707002

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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.
An entanglement purification protocol (EPP) aims to transform multiple noisy entangled states into a single entangled state with higher fidelity. In this work we consider input-independent EPPs that always yield an output fidelity no worse than each of the original noisy states, a property we call universality. We prove there is no n-to-1 EPP implementable by local operations and classical communication that is universal for all two-qubit entangled states, whereas such an EPP is possible using more general positive partial transpose-preserving (PPT) operations. We also show that universality is impossible by any bilocal Clifford EPP even when restricted to states with fidelities above an arbitrarily high threshold.
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.
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.
Abstract Quantum communication enables secure information transmission and entanglement distribution, but these tasks are fundamentally limited by the capacities of quantum channels. While quantum repeaters can mitigate losses and noise, entanglement swapping via a central node is ineffective against the Pauli dephasing channel due to degradation from Bell-state measurements. This suggests that purifying distributed Bell states before entanglement swapping is necessary. Although one-way hashing codes are known to saturate the dephasing channel capacity, no explicit two-way purification protocol has previously been shown to achieve this bound. In this work, we present a two-way entanglement purification protocol with an explicit, scalable circuit that asymptotically achieves the dephasing channel capacity. With each iteration, the fidelity of Bell states increases. At the final round, the residual dephasing error is suppressed doubly-exponentially, scaling as O ( p 2 n ) , enabling near-perfect Bell pairs for any fixed number of purification rounds n . The explicit circuit we propose is versatile and applicable to any number of Bell pairs, offering a practical solution for mitigating decoherence in quantum networks and distributed quantum computing.
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.
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.