C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. Wooters, "Mixed state entanglement and quantum error correcting codes," quant-ph/9604024.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of quantum error correcting codes. In fact, the code subspace for an additive code can be described as the space of ground states of the Hamiltonian formed by the sum of the stabilizers [22] For example, the code subspace for a 5 qubit code [5,1,3] encoding 1 qubit against single bit errors [23,24] is the space of ground states of the translation invariant Hamiltonian on a one dimensional lattice of five qubits: H [5,1,3] X j Z j 1 Z j 2 X j 3 , where the subscripts are to be interpreted mod 5. The space of ground states is two dimensional which is why it can encode 1 qubit. A ....

C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wootters, "Mixed-state entanglement and quantum error correction", Phys. Rev. A 54 (1996) 3824--3851.

....need in general not be eigenvectors of #, i.e. they need not be orthogonal. In order to obtain a well defined measure of entanglement, we can look for the optimal decomposition of #, which yields the smallest average entanglement (10.8) The latter is known as entanglement of formation E F [111]. Apart from this rather abstract definition, E F can also be interpreted in a more intuitive way: it is closely connected to the maximal number N of qubit pairs in the given state #, which can be produced by local operations (possibly with classical communication between the two parties ) if ....

.... if # remains bounded or not) 10.3 Does the optimal LSD yield a measure of entanglement Furthermore, Theorem 2 provides a connection between the BSA and the concurrence of #, which was originally [126] introduced as an auxiliary quantity in order to calculate the entanglement of formation E F [111]. Apart from the explicit formula (10.11) the concurrence of a mixed state is defined (similarly to E F ) as the minimum of the average concurrence #c# p i c(# i )overall decompositions # = i p i of # into pure states. After decomposing # s into product states, also the optimal ....

C. H. Bennet, D. P. DiVincenzo, J. Smolin, and W. K. Wootters, MixedState Entanglement and Quantum Error Correction,Phys.Rev.A54, 3824 (1996).

....on a system s can be thought of as orthogonal measurements on an extended system , which may not be orthogonal in s alone. These have applications in a broad number of areas including precision measurement [79] quantum communication in the context of entanglement 111 purification [80], and quantum error correction [22] The existence of POVMs that optimize information retrieval and state processing in various contexts have been discovered, but it remains a problem to implement them in real physical systems. Here we study systematic constructions of generalized measurements on ....

C. H. Bennett, et al., "Mixed State Entanglement and Quantum Error Correction", Phys. Rev. A 54, 3824 (1996).

....D, Pi C g Pi C Pi C : Let S be another set of operators. Then C is an S correcting code iff it is an S S detecting code . The idea of focusing on detectability rather than S = fA B j A; B 2 Sg. correctability and the relationship between the two notions is due to Bennett et al. [1]. The formalization in terms of projection operators is an immediate consequence of the conditions given in [7, 1] In the case where E = 0 , an interesting problem is to find codes which can correct all errors involving at most e of the factors of the underlying space. In that case S e is ....

....S detecting code . The idea of focusing on detectability rather than S = fA B j A; B 2 Sg. correctability and the relationship between the two notions is due to Bennett et al. 1] The formalization in terms of projection operators is an immediate consequence of the conditions given in [7, 1]. In the case where E = 0 , an interesting problem is to find codes which can correct all errors involving at most e of the factors of the underlying space. In that case S e is taken to consist of tensor products of error operators in E 0 with at most e of the factors not the identity. An ....

C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters. Mixed state entanglement and quantum error-correcting codes. quantph /9604024, 1996.

....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, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Mixed-state entanglement and quantum error correction. In Physical Review A, vol. 54, No. 5, November 1996.

....we can again confine ourselves to them. It is always possible to determine operators e 1 ; e 2 ; e p 2m in such a way that one of them, say e 1 , is the identity operator I p m . Define the weight of E in (2) as wt(E) jfoe i 6= I p m gj: 3) In the depolarizing channel model of errors [4], the operators e 2 ; e 3 ; satisfy Tr(e y i e j ) p m ffi i;j , where Tr is the trace of linear operators. When transmitting a qubit through a depolarizing channel, the probability that it is untouched (i.e. affected by the identity operator) is 1 Gammar and the probability that it is ....

C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wootters, "Mixed state entanglement and quantum error-correcting codes," Phys. Rev. A, vol. 54, pp. 3824--, 1996.

....are distinct elements of B, then hu wju wi = 0, hence 0 = hu wjEju wi = hujEjui hwjEjwi: 6 1. CLIFFORD CODES Therefore, hujEjui = hwjEjwi for all u; w 2 B. It follows that (1.4) holds for arbitrary u; w 2 Q with kuk = kwk. It has been shown by Knill and La amme [17] see also Bennett et al. [4]) that this error correction condition is not only necessary but also sucient. Thus, to summarize, a set of errors S can be corrected by a quantum error control code Q if and only if all errors in the set S y S = fE y 1 E 2 j E 1 ; E 2 2 Sg are detectable by Q. An elementary proof of this ....

....by Q. A quantum error control code with minium distance d = 2t 1 allows to correct decoherence errors a ecting up to t qudits. REMARKS (a) A detailed analysis of general quantum error control codes can be found in Knill and La amme [17] Another early account is given by Bennett et al. [4]. We refer to articles by Knill, La amme, and Viola [18] and by Zanardi [25] for more recent discussions of the general theory of quantum error control codes. b) The notion of detectable errors has been explicitly introduced in [17] in the form (1.6) The equivalent form (1.4) has been used by ....

C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. Wootters. Mixed state entanglement and quantum error correction. Physical Review A, 54:3824-3851, 1996.

..... Error algebra: E # # . E 2 = E 1 E 1 # E 1 # E 0 = span I Q . Definition: A (c q) code has minimum distance d for E 1 if it detects E d 1 . Theorem 11: A minimum distance 2e 1 (c q) code can be used to correct up to e errors (any error in E e ) Bennett al. 1996 [8], Knill Laflamme 1996 [9] Knill Laflamme Viola 1999 [6] General reference: M)ike 2001 [10] 20 TOC Group Codes I . G A group. # : G # MatN (C) # : g ## # g Unitary, faithful irrep. Subspaces via irreps of subgroups: Let H # G be a subgroup, # i irreps occurring in # # H. ....

C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters. Mixed state entanglement and quantum error-correcting codes. Phys. Rev. A, 54:3824--3851, 1996.

....constant is necessary so that the definition applies to trace decreasing operations. In what follows, we will see that is independent of ae C . An operation is reversible on a subspace C if there is an operation R which reverses it on this subspace. Knill and Laflamme [47] and Bennett et al. [48] derived necessary and sufficient conditions for the reversibility of an operation A on a subspace or code C. For an arbitrary decomposition fA j g of the operation and any basis fjaig for the code subspace, the conditions are: hajA y k A j jbi = m jk ffi ab (6.20) That is, when orthogonal ....

....of capacity has arisen in quantum mechanics, depending, for example, on whether the entanglement of a system with some reference system is required to be preserved by the transmission process, or not. Here we concentrate on two notions of quantum capacity, one investigated for example in [48], 51] 50] 60] and concerned with the maximum size of a Hilbert space all of whose pure states can be preserved with high fidelity, and another arising for example in [16] 17] 61] 19] concerned with the maximum entropy of a density operator whose entanglement with a reference system which ....

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C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, "Mixed state entanglement and quantum error correction," Physical Review A, vol. 54, pp. 3824, 1996.

.... two notions of quantum capacity, one investigated for example in [13] 14] 15] 16] concerned with the maximum entropy of a density operator whose entanglement with a reference system which does not undergo the noise process can be preserved with high delity, and another arising for example in [17], 18] 19] 20] and concerned with the maximum H. Barnum is with the School of Natural Science and ISIS, Hampshire College, Amherst, MA 01002. E mail: hbarnum hampshire.edu. E. Knill is with the Los Alamos National Laboratories, Mail Stop B265, Los Alamos, NM 87545. E mail: knill lanl.gov. M. ....

.... many authors have worked on the problem of quantum information transmission through quantum channels; some of this work calculates or or bounds the capacity we study here, for particular channels or classes of channels: an incomplete list that could serve as an entry to the literature includes [17], 23] 20] 18] 19] 7] 9] 24] Some of the extensive literature on the more algebraic approach to quantum coding also yields information about the quantum capacity. II. Quantum sources and channel capacity A. Mathematical preliminaries and notation The e ect of encoding procedures, ....

[Article contains additional citation context not shown here]

C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, \Mixed state entanglement and quantum error correction," Phys. Rev. A, vol. 54, pp. 3824, 1996.

....quantum state sys can be expressed only in terms of the initial state sys of the system and some interaction operators which completely specify the channel. 3.2. Depolarizing and Erasure Channel To illustrate the preceding, we consider two important quantum channels. Over a depolarizing channel [4], quantum information is transmitted undisturbed with probability , and it is replaced by a completely randomized quantum state with probability . A common assumption is that errors act independently on each qubit. In this case, for a single qubit equation (1) reads sys sys sys sys (where , ....

Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters, "Mixed State Entanglement and Quantum Error Correction", Physical Review A, vol. 54, no. 5, pp. 3824--3851, Nov. 1996.

.... in the development of error correction schemes for quantum information systems [1 10] This includes methods for converting classical error correcting codes to into quantum error correcting codes [2,3] formalizations of necessary and sufficient conditions for sets of states to form quantum codes [11,7,12], and a mathematical framework for a large class of quantum codes, known as stabilizer codes [8,9] In order to actually use quantum codes in quantum information systems, constructive methods for performing encodings, error correction, and decodings are required. Towards this end, gate arrays that ....

.... producing gate arrays that perform error correction for any stabilizer code have been presented [13] For computing encodings, the only gate arrays that have been proposed apply either to one specific code (such as one that encodes one qubit as five qubits and protects against a onequbit error) [4,7,14], or to restricted classes of stabilizer codes [3,10] In the present paper, we show how to efficiently construct a gate array that computes encodings for any stabilizer code. In the case of an n qubit code defined in terms of d generators, our gate array consists of at most nd operations (which ....

C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. Wooters, "Mixed state entanglement and quantum error correcting codes," quant-ph/9604024.

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C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. Wooters, "Mixed state entanglement and quantum error correcting codes," quant-ph/9604024.

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C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. Wooters, "Mixed state entanglement and quantum error correcting codes," Phys. Rev. A Vol. 54, No. 5, pp. 3824--3851 (1996).

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C. H. Bennett, D. DiVincenzo, J. A. Smolin, and W. K. Wootters, "Mixed state entanglement and quantum error correction," Phys. Rev. A, vol. 54, pp. 3824--3851, 1996; also LANL eprint quant-ph/9604024.

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C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters. Mixed state entanglement and quantum error correction. Phys. Rev. A, 54(5):3824-3851, 1996. quant-ph/9604024. 38

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C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. Wootters. Mixed state entanglement and quantum error correction. Phys. Rev. A, 54:3824-- 3851, 1996.

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C. Bennett et al., "Mixed State Entanglement and Quantum Error Correction," PRA 54, p. 3824, 1996, quant-ph/9604024.

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Bennett C H, DiVincenzo D P, Smolin J A and Wootters W K 1996 Mixed-state entanglement and quantum error correction Phys. Rev. A 54 3824--51 (Preprint quant-ph/ 9604024)

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C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. Wootters, Mixed State Entanglement and Quantum Error Correction, Phys. Rev. A 54, 3824-3851 (1996).

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Bennett C H, DiVincenzo D P, Smolin J A and Wootters W K Mixed state entanglement and quantum error correction, Phys. Rev. A 54 3825, 1996

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Bennett, C. H., DiVincenzo, D. P., Smolin, J. A., Wootters, W. K., Mixed state entanglement and quantum error correction. Phys. Rev. A 54, 3824{ 3851 (1996).

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Bennett, C. H., DiVincenzo, D. P., Smolin, J. A., Wootters, W. K., Mixed state entanglement and quantum error correction. Phys. Rev. A 54, 3824-- 3851 (1996).

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Bennett C H, DiVincenzo D P, Smolin J A and Wootters W K Mixed state entanglement and quantum error correction, Phys. Rev. A 54 3825, 1996

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C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Mixed-state entanglement and quantum error correction. In Physical Review A, vol. 54, No. 5, November 1996.

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