C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, "Purification of noisy entanglement and faithful teleportation via noisy channels," Phys. Rev. Lett., vol. 76, pp. 722--725, 1996; also LANL eprint quant-ph/9511027.  Home/Search   Document Not in Database   Summary   APS   Related Articles   Check

....(Chapter 5) 4.3.1. Distillation of pairs of qublts. Concrete distillation protocols are in general rather complicated procedures. We will sketch in the following how any pair of entangled qubits can be distilled. The first step is a scheme proposed for the first time by Bennett et al. [12]. It can be applied if the maximally entangled fraction (Equation (3.4) is greater than 1 2. As indicated above, we assume that Alice and Bob share a large amount of pairs in the state p, so that the total state is pN. To obtain a smaller number of pairs with a higher they proceed as follows: ....

.... rates [133] The first measure we want to present is oriented along the discussion of pure states: We define, roughly speaking, the asymptotic rate with which maximally entangled qubits can be distilled at most out of a state p g(J( K) as the Entanglement of Distillation ED(p) of p; cf [12]. To be more precise consider all possible distillation protocols for p (cf. Section 4.3) i.e. all sequences of LOCC channels TN: aN C aN ) N X (5.7) lim IIT (p IN) NI IIx = 0 (5.8) holds with a sequence of maximally entangled states f C aN . Now we can define lo82 (d) ED(p) sup limsup , ....

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin and W. K. Wootters. Purification of noisy entanglement and faithful teleportation via noisy channels. Phys. Rev. Lett. 76, no. 5, 722-725 (1996). Erratum: Phys. Rev. Lett. 78, 10, 2031 (1997).

....has recently been seen in the context of private quantum channels [11] These examples serve as a caution to the entire quantum information community when dealing with approximate performance of quantum protocols. The idea of using a randomized encoding algorithm is not new in QECC. In particular [5] have devised codes that can correct more (malicious) errors on average than any deterministic QECC. However, their model significantly differs from ours in one of two ways: they assume either that the errors occur at random or that the code is randomly agreed by the coder and the decoder but is ....

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, "Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels.", Phys. Rev. Lett. 76 (1996) 722-725, Quantum Physics, abstract quant-ph/9511027.

....to Bob, authenticating the 2m classical bits transmitted in the teleportation protocol. If initially share only a classical key, however, the task is more difficult. One approach is to first distribute EPR pairs (which might get corrupted in transit) and then use entanglement purification [6] to establish clean pairs for teleportation. This can be improved: we do not need a full scale entanglement purification protocol, which produces good EPR pairs even if the channel is noisy; instead we only need something we call a purity testing protocol, which checks that EPR pairs are ....

....Measuring the syndrome projects a quantum state into one of these codes. Actually, the standard coset al..so depends on the selection of a basis of generators for S. 2. 3 Purification and purity testing QECC s may be used for a task known as entanglement purification (first defined in [6]) In this setting, share some Bell states (say 00# 11#) which have been corrupted by transmission through a noisy quantum channel. They want a protocol which processes these imperfect EPR pairs and produces a smaller number of higher quality pairs. We assume that have access ....

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, W. K. Wootters, "Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels", Phys. Rev. Lett., vol. 76, 1996, pp. 722 -- 725.

....the focus of much research; we list the most relevant works here. Bennett et al. 4] gave a protocol for the case that Alice and Bob share identical copies of the pure state j i = cos j 01i sin j 10i) This was extended to the case when Alice and Bob share identical copies of a mixed state [5, 6, 11]. Vidal [24] and subsequently, Jonathan and Plenio [12] Hardy [10] and Vidal, Jonathan, and Nielsen [25] considered extracting entanglement from a single copy of an arbitrary pure state, assuming that we know a complete description of the state. All these works use relatively simple models for ....

C. H. Bennett, H. J. Bernstein, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels, In Physics Review Letters, vol. 76, page 722, 1996.

....the 2m classical bits transmitted in the teleportation protocol. If A and B initially share only a classical key, however, the task is more difficult. We start with a simple approach: first distribute EPR pairs (which might get corrupted in transit) and then use entanglement purification [7] to establish clean pairs for teleportation. This can be improved: we do not need a full scale entanglement purification protocol, which produces good EPR pairs even if the channel is noisy; instead we only need something we call a purity testing protocol, which checks that EPR pairs are correct, ....

....as [16] Since we rely heavily on terminology and techniques from quantum error correction (especially stabilizer codes) appendix A provides a summary of the relevant notions. 2. 2 Purification and purity testing Quantum error correcting codes (QECCs) may be used for entanglement purification ([7]) In this setting, A and B share some Bell states (say j i = j00i j11i) which have been corrupted by transmission through a noisy quantum channel. They want a protocol which processes these imperfect EPR pairs and produces a smaller number of higher quality pairs. We assume that A and B have ....

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, W. K. Wootters, "Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels", Phys. Rev. Lett., vol. 76, 1996, pp. 722 -- 725.

....Key Distribution protocol of Lo and Chau [10] as modified by Shor and Preskill [14] If interaction is allowed between A and B then A may send B halves of singlet states. Then a quality test may be run to determine the accuracy of these singlets. From this estimate, quantum purification techniques [6] may be used to obtain fewer almost pure singlets. In turn, these can be used for teleportation as needed for the above scheme. However, we do not want of an interactive solution. If we wish to store authenticated quantum states and only later verify their authenticity, interaction is absolutely ....

....C 2 and C 1 , and encoding and decoding states from within the spaces Q s 1 ;s 2 , can be quite done efficiently given only generator matrices for C 2 and C 1 . Purification and purity testing It turns out that QECC s may be used for a task known as EPR pair purification (first defined in [6]) In this setting, A and B share some Bell states (say j i = j00i j11i) which have been corrupted by transmission through a noisy quantum channel. They want a protocol whch provides some guarantee of the purity of the resulting states. We assume that they have access to an authenticated, ....

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, W. K. Wootters, "Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels", Phys. Rev. Lett., vol. 76, 1996, pp. 722 -- 725

....Finally, Alice and Bob measure their maximally entangled systems and establish a secret key. This way of obtaining a key directly from a quantum state Psi , without any error correction nor classical privacy amplification, is called quantum privacy amplification 3 (QPA for short) 8] [2]. Note that the procedure described in [8] and [2] guarantees that Eve s relative information (relative to the key length) is arbitrarily small, but not that her absolute information is negligible. The analog of this problem in the classical case is discussed in [21] The precise conditions under ....

....entangled systems and establish a secret key. This way of obtaining a key directly from a quantum state Psi , without any error correction nor classical privacy amplification, is called quantum privacy amplification 3 (QPA for short) 8] 2] Note that the procedure described in [8] and [2] guarantees that Eve s relative information (relative to the key length) is arbitrarily small, but not that her absolute information is negligible. The analog of this problem in the classical case is discussed in [21] The precise conditions under which a general state ae AB can be purified are ....

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wooters, Purification of noisy entanglement and faithful teleportation via noisy channels, Phys. Rev. Lett., Vol. 76, pp. 722--725, 1996.

....quantum information by teleportation, the parties need EPR pairs. If these EPR pairs are distributed over noisy quantum channels, the teleportation process is also noisy. It has been shown that it is possible to distill a small number of better EPR pairs starting from a supply of many EPR pairs [21]. This process of entanglement purification is sketched as follows. Both parties operate locally on their half of each EPR pair. Some of the qubits are measured, and the results of the measurement are communicated. Hence, in addition to the quantum channel for the distribution of the EPR pairs, a ....

Charles H. Bennett, Gilles Brassard, Sandu Popescu, Benjamin Schumacher, John A. Smolin, and William K. Wootters, "Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels", Physical Review Letters, vol. 76, no. 5, pp. 722--725, 28. Jan. 1996.

....for an entangled state. Buhrman et al. [BCW98] Adami, Cerf [AC98b] contrast quantum information theory with classical information theory. Quantum channel capacity has been investigated for noisy channels (DiVincenzo, et al. [DSS 95] Holevo [Hol96] Barnum et al. [BNS 97] Bennett et al. [BDS98,BBP 96] very noisy channels (Shor, Smolin [SS98] and quantum erasure channels (Bennett et al. [BDS97b] Fuchs [Fuc97] showed that nonorthogonal quantum states maximize classical information capacity. Also, Helstrom [H97,H98] defines a quantum theory of information detection, and Fuchs [Fuc96] ....

....distant nodes. For example, CZ97] use a cavity QED device that traps atoms in multiple cavities and exchanges photons between the cavities to establish the distributed entanglement. Various basic difficulties were overcome: How can one do state transfer distribution Bennett et al. [BBC93, BBP 96] Brassard [Bra96] developed a technique known as teleportation to transmit arbitrary input states with perfect fidelity. It does this by separating the input state into classical and quantum components. The input can then be reconstructed from these components with perfect fidelity. How can ....

Bennett, C. H., Brassard, Popescu, Schumacher, Smolin, and Wootters, Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels, Phys. Rev. Lett. 76, 722 (1996).

....that quantum bits cannot be cloned [Peres 1993: Section 9 4] but this argument does not rule out more complicated ways of reducing inaccuracy or decoherence using software. In fact, some progress in the direction of reducing inaccuracy [Berthiaume et al. 1994] has already been made. The result of Bennett et al. 1995] that quantum bits can be faithfully transmitted over a noisy quantum channel gives further hope that quantum computations can similarly be faithfully carried out using noisy quantum bits and noisy quantum gates. Discrete logarithms and factoring are not in themselves widely useful problems. They ....

C. H. Bennett, G. Brassard, B. Schumacher, J. Smolin and W. K. Wooters (1995) "Purification of noisy entanglement, and faithful teleportation via noisy channels," preprint.

....after the required preparation. Thus, although working prototypes of quantum teleportation have recently been demonstrated [8,9] quantum teleportation across significant time and space will have to await a technology that allows for the efficient long term storage [28,17,30] and purification [6,7] of quantum information. Nevertheless, it may be that shortdistance quantum teleportation will play a role in transporting quantum information inside quantum computers. Thus we see that the fates of quantum computing and quantum teleportation are inseparably entangled 2 Quantum teleportation ....

Bennett, Charles H., Gilles Brassard, Sandu Popescu, Benjamin Schumacher, John A. Smolin and William K. Wootters, "Purification of noisy entanglement and faithful teleportation via noisy channels", Physical Review Letters, Vol. 76, no. 5, 29 January 1996, pp. 722--725.

....it, so because of the theorem that a quantum bit cannot be cloned, it was widely believed that these techniques could not be applied to quantum information. That quantum error correcting codes could indeed exist was recently shown by one of us [32] At around the same time, Bennett et al. [2] discovered that two experimenters each holding one component of many noisy Einstein Podolsky Rosen (EPR) pairs could purify them, yielding fewer nearly perfect EPR pairs. The resulting pairs can then be used to teleport quantum information from one experimenter to the other [1] These two ....

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin and W. K. Wooters, "Purification of noisy entanglement and faithful teleportation via noisy channels," Phys. Rev. Lett., 76, pp. 722--725 (1996).

....be compared with theoretical upper bounds of min h 1 Gamma H 2 (2t=3n) H 2 i 1 2 q (1 Gamma t=n)t=n ji for t=n 1 2 , and 0 for t=n 1 2 . These are obtained from bounds on the quantum information capacity of a quantum channel, which we derive in Section V from results of Refs. [10, 11]. These bounds are plotted in Fig. 1 in Section V. II. DEFINITIONS Our constructions of quantum error correcting codes rely heavily on the properties of classical error correcting codes. We will thus first briefly review certain definitions and properties related to binary linear error correcting ....

....the average behavior of the channel results in the decoherence of fewer than t qubits, a channel may still be able to transmit quantum states very well. A measure of the success of transmission of quantum states that has previously been successful applied in quantum information theory is fidelity [15, 11]. In this paper, we define fidelity slightly differently from the definition in Refs. 15] we make this change as these previous papers discuss channels that transmit some distribution of quantum states given a priori, whereas we want our channel to faithfully transmit any pure input state. ....

[Article contains additional citation context not shown here]

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wooters, "Purification of noisy entanglement and faithful teleportation via noisy channels," Phys. Rev. Lett. (to appear).

....showed that a class of good quantum codes could be obtained by using a construction that starts with a binary linear code C containing its dual C . Independently, Steane also discovered the existence of quantum codes [73] and the same construction [72] At around the same time, Bennett et al. [4] discovered that two experimenters each holding one component of many noisy Einstein Podolsky Rosen (EPR) pairs could purify them using only a classical channel to obtain fewer nearly perfect EPR pairs. The resulting pairs can then be used to teleport quantum information from one experimenter to ....

.... codes by Gottesman [37] However, the most efficient methods currently known for fault tolerant computation [2] 44] 48] 75] use only Calderbank Shor Steane codes (cf. Theorem 9) iv) It turns out that the proofs of the lower bounds on the capacity of quantum channels given in Bennett et al. [4], 5] and DiVincenzo, Shor and Smolin [31] can be restated in terms of additive codes. In particular, this implies that these bounds can be attained using additive codes. v) Cleve [20] has found a way to apply asymptotic upper bounds for classical binary codes to additive codes. vi) Steane [74] ....

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin and W. K. Wootters, "Purification of noisy entanglement and faithful teleportation via noisy channels," Phys. Rev. Lett., 76, pp. 722--725 (1996); also LANL e-print quant-ph/9511027.

....for an entangled state. Buhrman et al. [BCW98] Adami, Cerf [AC98b] contrast quantum information theory with classical information theory. Quantum channel capacity has been investigated for noisy channels (DiVincenzo, et al. [DSS 95] Holevo [Hol96] Barnum et al. [BNS 97] Bennett et al. [BDS98,BBP 96] very noisy channels (Shor, Smolin [SS98] and quantum erasure channels (Bennett et al. [BDS97b] Fuchs [Fuc97] showed that nonorthogonal quantum states maximize classical information capacity. Also, Helstrom [H97,H98] defines a quantum theory of information detection, and Fuchs [Fuc96] ....

....distant nodes. For example, CZ97] use a cavity QED device that traps atoms in multiple cavities and exchanges photons between the cavities to establish the distributed entanglement. Various basic difficulties were overcome: How can one do state transfer distribution Bennett et al. [BBC93, BBP 96] Brassard [Bra96] developed a technique known as teleportation to transmit arbitrary input states with perfect fidelity. It does this by separating the input state into classical and quantum components. The input can then be reconstructed from these components with perfect fidelity. How can ....

Bennett, C. H., Brassard, Popescu, Schumacher, Smolin, and Wootters, Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels, Phys. Rev. Lett. 76, 722 (1996).

....David Deutsch, Artur K. Ekert, Richard Jozsa, Chiara Macchiavello, Sandu Popescu and Anna Sanpera in Oxford [32] proved wrong earlier claims that the use of nonlocality held no significant benefit over the original BB84 protocol [11] In particular, the use of entanglement purification techniques [12] yields a protocol that has no analogue along the lines of BB84. In brief, sender and receiver exchange entanglement through a noisy and possibly bugged quantum channel. Because of the potential eavesdropper and also because of natural noise, the resulting entanglement is imperfect. Using ....

Bennett, C. H., G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin and W. K. Wootters, "Purification of noisy entanglement and faithful teleportation via noisy channels" Physical Review Letters, Vol. 76, no. 5, 29 January 1996, pp. 722 -- 725.

No context found.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, "Purification of noisy entanglement and faithful teleportation via noisy channels," Phys. Rev. Lett., vol. 76, pp. 722--725, 1996; also LANL eprint quant-ph/9511027.

No context found.

C.H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J.A. Smolin, and W.K. Wootters. Purification of noisy entanglement and faithful teleportation via noisy channels. Phys. Rev. Lett., 76:722--725, 1996.

No context found.

Bennett C H, Brassard G, Popescu S, Schumacher B, Smolin J A and Wootters W K 1996 Purification of noisy entanglement and faithful teleportation via noisy channels Phys. Rev. Lett. 76 722--5 (Preprint quantph / 9511027)

No context found.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, W. K. Wootters, "Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels ", Phys. Rev. Lett., vol. 76, 1996, pp. 722 -- 725.

No context found.

C.H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J.A. Smolin, and W.K. Wootters, Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels, Phys. Rev. Lett. 76, 722-725 (1996).

No context found.

C. H. Bennett, H. J. Bernstein, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels, In Physics Review Letters, vol. 76, page 722, 1996. 9

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C.H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J.A. Smolin, and W.K. Wootters: "Purification of noisy entanglement and faithful teleportation via noisy channels", Phys.Rev.Lett. 76(1996) 722--725

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