B. Schumacher, "Quantum coding" Phys. Rev. A 51, 2738 (1995).  Home/Search   Document Details and Download   Summary   APS   Related Articles   Check

....that the actual global (perhaps common) minimax and maximin are achieved for probability distributions not belonging to the one parameter family q u . In analogy to [17, Sec. 5. 2] the matrices n (u) should prove useful for the universal version of Schumacher data compression [7] 19] 30] [47]. Schumacher s result [47] 30] must be considered as the quantum analogue of Shannon s noiseless coding theorem (see e.g. 60, Sec. 5.6] Roughly, quantum data compression, as proposed by Schumacher [47] works as follows: A (quantum) signal source ( sender ) generates signal states of a ....

....(perhaps common) minimax and maximin are achieved for probability distributions not belonging to the one parameter family q u . In analogy to [17, Sec. 5. 2] the matrices n (u) should prove useful for the universal version of Schumacher data compression [7] 19] 30] 47] Schumacher s result [47], 30] must be considered as the quantum analogue of Shannon s noiseless coding theorem (see e.g. 60, Sec. 5.6] Roughly, quantum data compression, as proposed by Schumacher [47] works as follows: A (quantum) signal source ( sender ) generates signal states of a quantum system M , the ....

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B. Schumacher, \Quantum coding," Phys. Rev. A, vol. 51, no. 4, pp. 2738-2747, April 1995.

....possibility that the encoder knows the identity of the state being encoded, sometimes called preparation visible encoding. The converse portion of the theorem is thus stronger than it would be if encodings were restricted to be completely positive maps, as it was in Schumacher s original proof [41]. The encoding used by Jozsa and Schumacher [15] in showing achievability is, however, a completely positive map. Thus, allowing preparation visible encoding does not increase capacity over the case when one is restricted to encoding via a completely positive map. Jozsa and Schumacher established ....

....S ( Sigma) is the entropy rate of a general source Sigma = fae (1) ae (n) g: S ( Sigma) j lim sup n 1 S(ae (n) s ) n : 5.7) The definition derives its interest from the fact that for some interesting sources for example, the i.i.d. source with ae (n) ae Omega n [41] all but a negligible portion of the source becomes concentrated in an ffl typical subspace as n goes to infinity, no matter how small ffl is chosen to be. More formally, the i.i.d. source satisfies the Quantum Asymptotic Equipartition Property (QAEP) Here and elsewhere, we will sometimes use ....

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B. Schumacher, "Quantum coding," Physical Review A, vol. 51, pp. 2738--2747, 1995.

....eigenvectors j i of (n) whose eigenvalues satisfy: 2 n(S( 2 n(S( 19) An equivalent requirement is: j 1 n log S( j : 20) The de nition derives its interest from the fact that for some interesting sources for example, the i.i.d. source with (n) n [29] all but a negligible portion of the source becomes concentrated in an typical subspace as n goes to in nity, no matter how small is chosen to be. More formally, the i.i.d. source satis es the Quantum Asymptotic Equipartition Property (QAEP) Here and elsewhere, we will sometimes use the ....

B. Schumacher, \Quantum coding," Phys. Rev. A, vol. 51, pp. 2738-2747, 1995.

....Our approach to capacity is based on quantum mutual information which is de ned in terms of relative entropy (or informational divergence) Therefore, relative entropy is the basic tool in the paper. The capacity is not compared with performance bounds of classical coding (as in [6] and [14]) because we are mainly interested in the purely quantum part of the channel. Section 2 contains some generalities about quantum communication channels, mutual information and relative entropy. Holevo s bound is also discussed and we show that it is rarely achievable. In Section 3 our capacities ....

B. Schumacher, Quantum coding, Phys. Rev. A (1995), 2738-2747

....channel coding theorem for quantum channels, and its strong converse. The technique is largely inspired by Wolfowitz s combinatorial approach using types of sequences. 1 Introduction After the recent achievements in quantum information theory, most notably Schumacher s quantum data compression [16], and the determination of the quantum channel capacity by Holevo [12] and independently by Schumacher and Westmoreland [17] building on ideas of Hausladen et.al. 7] we feel that one should try to convert other and stronger techniques of classical information theory than those used in the ....

....between the various sequences (see note after lemma 10) For 0 j 1 a positive operator B on H with B 1 will be called an j image of A ae X if for all a 2 A: Tr (W a B) j. The size of the operator B is defined as Tr B, 1 Distinguish from the typical projector introduced by Schumacher [16] (which we would like to call entropy typical ) though there are similarities, as will be seen below. 4 and g W (A; j) is the minimal size of j images of A. The following basic results are in the classical situation from [3] Lemma 2 (Type counting) There are at most (n 1) jX j types of ....

B. Schumacher, "Quantum Coding", Phys. Rev. A 51,4(1995), 2738--2747 17

....see that S(ae) S X i p i jOE i ihOE i j = H(p) where H(p) is the Shannon entropy of the probability distribution p 1 ; p 2 ; A source E = f(ae i ; p i )g has an associated Von Neumann entropy S(ae) of the average state ae = P i p i ae i . Schumacher s noiseless coding theorem [16] shows how to obtain an encoding with average letter length S(ae) for a source of pure states, where the fidelity of the encoding goes to 1 as the number of letters emitted by the source goes to infinity. A survey can be found in Preskill s lecture notes [15, page 190] or in Nielsen s thesis [14, ....

Benjamin Schumacher, "Quantum Coding", Physical Review A, Volume 51, No. 4, pp. 2738--2747 (1995)

....arithmetic coding, and Lempel Ziv coding for achieving the Shannon entropy limit are known in the classical case; see, for example, Cover and Thomas [10] In comparison, the field of quantum data compression is still nascent. The quantum analogue to Shannon s theorem is Schumacher s theorem [11]: if the per symbol code rate is slightly larger than the von Neumann entropy, then there exists a block code (with sufficiently large block size) such that the compressed message can be recovered with average fidelity close to unity. The similarity of the two theorems makes it possible to use, to ....

....2 matrix given by the outer product between the vector jOEi and its conjugate transpose hOEj. The problem of (pure state) quantum noiseless data compression is to the transmit such sequences of symbols with high fidelity, using a minimal number of quantum bits. According to Schumacher s theorem[11], on average each symbol can be transmitted in (slightly larger than) S(ae) Tr(ae log ae) quantum bits with high probability of correct reception, where S(ae) is known as the von Neumann entropy. A surprising contrast between the classical and the quantum cases is that S(ae) H(p) where ....

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B. Schumacher, "Quantum coding," Physical Review A, vol. 51, pp. 2738--2747, 1995.

.... created: 1998.08.13) On reliability function of quantum communication channel M. V. Burnashev , A. S. Holevo y 1. Introduction Let H be a Hilbert space. We consider quantum channel [4] with a nite input alphabet f1; ag and with pure signal states S i = j i i j; i = 1; a. Compound ....

.... created: 1998.08.13) On reliability function of quantum communication channel M. V. Burnashev , A. S. Holevo y 1. Introduction Let H be a Hilbert space. We consider quantum channel [4] with a nite input alphabet f1; ag and with pure signal states S i = j i i j; i = 1; a. Compound ....

[Article contains additional citation context not shown here]

B. Schumacher, Quantum coding, Phys. Rev. A 51, N 4, 2738-2747 (1995).

....quantum systems. We propose a form of universal coding for the situation in which the density matrix describing an ensemble of quantum signal states is unknown. A sequence of n signals would be projected onto the dominant eigenspaces of i n (u) 1. Introduction A theorem has recently been proven [30, 47] (cf. 7, 19, 35] in the context of quantum information theory [7, 40] that is analogous to the noiseless coding theorem of classical information theory. In the quantum result, the von Neumann entropy [39, 58] S(ae) Gamma Tr ae log ae (1.1) equalling the Shannon entropy of the probability ....

....obtain some of our results by making use of representation theory of SU(2) An even more general result was derived by combining these two approaches. We comment on this issue at the end of Sec. 3. The matrices i n (u) should prove useful for the universal version of Schumacher data compression [7, 19, 30, 47] by projecting blocks of n signals (qubits) onto those typical subspaces of 2 n dimensional Hilbert space corresponding to as many of the dominant eigenvalues of i n (u) as it takes to exceed a sum 1 Gamma ffl. This can be accomplished by a unitary transformation, the inverse of which would ....

B. Schumacher, "Quantum coding," Phys. Rev. A, vol. 51, no. 4, pp. 2738-2747, April 1995.

....without fanout or feedback by quantum wires . The gates have the same number of inputs as outputs, and a gate of n inputs represents a unitary operation of the group U(2 n ) i.e. a generalized rotation in a Hilbert space of dimension 2 n . Each wire represents a quantum bit, or qubit [25, 26], i.e. a quantum system with a 2 dimensional Hilbert space, capable of existing in a superposition of Boolean states and of being entangled with the states of other qubits. Where there is no danger of confusion, we will use the term bit in either the classical or quantum sense. Just as ....

B. Schumacher, "On Quantum Coding," Phys. Rev. A (in press to appear 1995).

....variables X i = Gamma log q Deltaji for the diagonalizations ae i = P j q jji ij . Observe that K may be any bound for the variance of the X i . 1 Where we identify (ae 1 ; ae n) with ae n = ae 1 Omega Delta Delta Delta Omega ae n . 2 This is essentially what Schumacher [15] calls typical subspace. We add the predicate entropy to distinguish it from another typicality concept introduced below. DRAFT January 22, 5 Motivated by this we define an j shadow of ae n to be a positive operator B 1 with Tr (ae n B) j. More generally an operator B as before is ....

.... Gamma F is immediate, and also the inequality 1 Gamma F D 2 (with the convexity of the square function) Observe that in the main text we only need lemma 2. But we believe that this presentation has its own value: for example it shows that the fidelity criterion in quantum coding [15] may be replaced by a k Delta k 1 criterion, which leads to a short proof of the converse, see [11] ....

B. Schumacher, "Quantum Coding", Phys. Rev. A 51,4 (1995), 2738--2747

....subspace. This possibility is allowed by the superposition principle of quantum mechanics but cannot occur in classical error correction. For realistic quantum computers only a subset of possible errors can be corrected. An appropriate measure of the quality of a recovered code is the fidelity [21]. Fidelity is the overlap between the final state ae f of a system ae and the original state j Psi i i. If the combined superoperator consisting of an interaction with the environment followed by a recovery operation is given by A = fA 0 ; g, then the fidelity is F (j Psi i i; A) h Psi i ....

Benjamin Schumacher. Quantum Coding. Phys. Rev. A, 51:2738, 1995.

....2, 6, 10, 11, 4] but no universal bound was proposed. The minimum time needed for a quantum system to pass from one orthogonal Supported by NSF grant DMS 9596217 and by DARPA contract DABT63 95 C 0130 1 Although many of the computing and communications properies of quantum systems are novel[14, 16, 3], basic questions about memory and speed seem to be rather independent of such considerations. state to another has also previously been characterized, in terms of DeltaE[7, 1, 15] This bound places no limit, however, on how fast a system with bounded average energy can evolve (since DeltaE ....

Schumacher, B., "Quantum Coding," Phys. Rev. A 51 (1995), 2738.

....there seems to be agreement that energy gets dissipated in the irreversible observation phase, to the author s knowledge it is not yet clear how much. This seems to require a quantum Kolmogorov complexity based on qubits (quantum bits) as defined in context of quantum information theory by [Schuhmacher, 1994], analogous to the classical bits of information theory of [Shannon, 1948] Through a sequence of proposals [Benioff, 1980 1986] Feynman, 1982 1987] Deutsch, 1985 1992] there has emerged a Turing machine model of quantum coherent computing. 4.1 Background: Probabilistic Turing Machines ....

, B.W. Schumacher, On Quantum coding, Phys. Rev. A, in press to appear in 1995; (with R. Josza), A new proof of the quantum noiseless coding theorem, J. Modern Optics, 41(1994), 2343-2349.

....in the most restrictive of these models, and prove the strong converse in the most powerful model. Keywords Quantum source, entanglement, fidelity, noiseless coding, strong converse. I. Introduction The subject of quantum information is very young, in fact one of its major roots is the paper [10] (which appeared only after its successor [8] It is concerned with the following basic problem of information theory: Consider a source emitting pure quantum states on a fixed complex Hilbert space H, say for definiteness from a finite set P, with probabilities P ( Suppose that we get the ....

....joint state n = 1 Omega Delta Delta Delta Omega n using as little resources as possible and such that the original state can be recovered from the encoding within a good approximation. The resource is to be basically the dimension of the Hilbert space of the encodings. As Schumacher [10] observed this gives an operational definition of the von Neumann entropy of the ensemble. In the following section we will precisely define the model(s) and discuss their relation, in section IV we will prove the noiseless coding theorem for the weakest model: quantum encoding together with ....

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B. Schumacher, "Quantum Coding", Phys. Rev. A 51,4 (1995), 2738--2747

....v4 12 Aug 1998 On reliability function of quantum communication channel M. V. Burnashev , A. S. Holevo y 1. Introduction Let H be a Hilbert space. We consider quantum channel [4] with a finite input alphabet f1; ag and with pure signal states S i = j i i j; i = 1; a. ....

.... v4 12 Aug 1998 On reliability function of quantum communication channel M. V. Burnashev , A. S. Holevo y 1. Introduction Let H be a Hilbert space. We consider quantum channel [4] with a finite input alphabet f1; ag and with pure signal states S i = j i i j; i = 1; a. Compound channel of ....

[Article contains additional citation context not shown here]

B. Schumacher, Quantum coding, Phys. Rev. A 51, N 4, 2738-2747 (1995).

....without fanout or feedback by quantum wires . The gates have the same number of inputs as outputs, and a gate of n inputs carries a unitary operation of the group U(2 n ) i.e. a generalized rotation in a Hilbert space of dimension 2 n . Each wire represents a quantum bit, or qubit [24, 25], i.e. a quantum system with a 2 dimensional Hilbert space, capable of existing in a superposition of Boolean states and of being entangled with the states of other qubits. Where there is no danger of confusion, we will use the term bit in either the classical or quantum sense. Just as ....

B. Schumacher, "On Quantum Coding," Phys. Rev. A (in press to appear 1995).

....In any case the asymptotic optimality of the bit extraction rate of our algorithm, and the near optimality of its runtime, imply that the only substantial remaining challenge is physical. Our initialization scheme may be thought of as carrying out an approximate form of Schumacher compression [22], a fundamental primitive in quantum information theory. Indeed, our initialization scheme can be substituted in place of Schumacher s scheme in many applications. The advantage of our scheme is that it requires no scratch bits, and is considerably more efficient than the most efficient known ....

B. Schumacher, "Quantum Coding", Phys. Rev. A. 51, 2738 (1995).

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B. Schumacher, "Quantum coding" Phys. Rev. A 51, 2738 (1995).

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B. Schumacher, "Quantum coding" Phys. Rev. A 51, 2738 (1995).

....perhaps ultimately practical) importance of questions about how quantum states can be compressed, transmitted across noisy or low dimensional channels, and recovered, and otherwise manipulated in a fashion analogous to classical information. Most of the work done on these matters, beginning with [1], has focused on the manipulation of pure states, with mixed states appearing only in intermediate stages, as the result of noise. An exception is [2] which considered the copying or broadcasting of mixed states. When mixed states have appeared as states to be transmitted, it has usually been ....

....or transmission. Indeed, quantum computation can enable more efficient than classical sampling from probability distributions [7, 8] there may be relations between these ideas and the work reported here. The problem of optimal compression for ensembles of pure quantum states has been solved [1, 9, 10], but for sources of mixed states the minimal resources are unknown. This question has also been considered by M. Horodecki in [11, 12] In this paper, we consider several variants of the question, depending on the fidelity criteria and encoding decoding procedures used. Sections 2 and 3 present ....

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B. W. Schumacher, "Quantum coding," Physical Review A, vol. 51, pp. 2738--2747, 1995.

....in coding, typically part of a system will be discarded in the process giving an evolution from a pure state to a mixed state. For perfect reconstitution we would have W i = ja i iha i j but in view of (b) this will not generally be the case. We define the fidelity F of the coding scheme to be [1] F = X i p i ha i jW i ja i i (1) 2 Recall that ha i jW i ja i i is the probability that the state W i passes the yes no test of being the state ja i i (where the test is the measurement of the observable proj ja i i ) A perfect coding scheme has W i = ja i iha i j and F = 1, but in general ....

....constitutes the entire information we have about the original state. If d dimensions per signal were used to code the states, then the W i s will be density matrices all supported in some fixed d dimensional subspace of H n . The quantum noiseless coding theorem, first introduced and proved in [1], solves the problem of the minimal resources necessary for (block) coding with arbitrarily high fidelity. By analogy with the classical measure of information, the bit, as a 2 state classical system, we use the term qubit [1] to refer to the quantum state storage capacity of a two dimensional ....

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B. Schumacher, "On Quantum Coding", Phys. Rev. A, (to appear 1993)

....Department of Mathematics and Statistics, University of Plymouth, Drake Circus, Plymouth, Devon PL48AA, U.K. b) Permanent address: Department of Physics, Technion Israel Institute of Technology, 32 000 Haifa, Israel. Recent schemes for quantum teleportation [1] and quantum noiseless coding [2, 3] exemplify a new approach to quantum communication theory. The issue is to determine what are the necessary and sufficient physical channel resources for transmitting quantum states accurately from a source to a receiver. This approach departs somewhat from the traditional emphasis on the ....

....of transmitting quantum states accurately without rendering them into classical form, the task of the decoding apparatus at the receiving end of the channel is to regenerate the original source states as accurately as possible. The accuracy of this regeneration is measured by its fidelity [2], F = X a p a h a jW a j a i; 3) where W a is the density matrix of the (in general mixed) output state resulting from source state a . When the output states are pure, the fidelity may be also defined by F = X a p a jh 0 a j a ij 2 : 4) where 0 a is the pure output state ....

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B. Schumacher, "On Quantum Coding," Phys. Rev. A (to appear 1993).

....i ih Psi i j: The von Neumann entropy corresponding to (S; p) is defined in terms of the density matrix ae as H VN (ae) GammaTr(ae log ae) In general, H VN (ae) H S (p) with equality occurring if and only if the states in S are mutually orthogonal. Roughly speaking, Schumacher s theorem [4] states that nH VN (ae) is asymptotically the maximum compression attainable for n qubits resulting from a source with density matrix ae. More precisely, let (S; p) be any ensemble of qubits, and ae be the corresponding density matrix. Then, for all ; ffi 0, for sufficiently large n and (n) ....

....if (n) n(H VN (ae) Gamma ffi) The proof of Schumacher s Theorem is based on the existence of a typical subspace of the Hilbert space of n qubits, which has the property that, with high probability, a sample of jff 1 ; ff n i has almost unit projection onto . It has been shown [4,5] that the dimension of is 2 nH VN (ae) thus, the operation that the compressor should perform involves transposing the subspace into the Hilbert space of a smaller block of nH VN (ae) qubits. Bennett [6] gives a more explicit procedure for accomplishing this transposition , which we ....

B. Schumacher, "Quantum coding" Phys. Rev. A 51, 2738 (1995).

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B. Schumacher. Quantum coding. Phys. Rev. A, 51:2738, 1995.

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