Asymptotic redundancies for universal quantum coding

Clarke and Barren (1990, 1994, 1995) have shown that the Jeffreys' invariant prior of Bayesian theory yields the common asymptotic (minimax and maximin) redundancy of universal data compression in a parametric setting. We seek a possible analog of this result for the two-level quantum systems. We restrict our considerations to prior probability distributions belonging to a certain one-parameter family, q/sub u/,-/spl infin/<u<1. Within this setting, we are able to compute exact redundancy formulas, for which we find the asymptotic limits. We compare our quantum asymptotic redundancy formulas to those derived by naively applying the (nonquantum) counterparts of Clarke and Barren, and find certain common features. Our results are based on formulas we obtain for the eigenvalues and eigenvectors of 2/sup n//spl times/2/sup n/ (Bayesian density) matrices, /spl zeta//sub n/(u). These matrices are the weighted averages (with respect to q/sub u/) of all possible tensor products of n identical 2/spl times/2 density matrices, representing the two-level 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 /spl zeta//sub n/(u).

[1]  David Haussler,et al.  A general minimax result for relative entropy , 1997, IEEE Trans. Inf. Theory.

[2]  Rangasami L. Kashyap,et al.  Prior probability and uncertainty , 1971, IEEE Trans. Inf. Theory.

[3]  Everett G. Larson,et al.  The Evolution of our Probability Image for the Spin Orientation of a Spin — 1/2 — Ensemble as Measurements are Made on Several Members of the Ensemble — Connections with Information Theory and Bayesian Statistics , 1991 .

[4]  T. Koornwinder,et al.  BASIC HYPERGEOMETRIC SERIES (Encyclopedia of Mathematics and its Applications) , 1991 .

[5]  L. Biedenharn.  Angular momentum in quantum physics , 1981 .

[6]  S. Adler,et al.  Quaternionic quantum mechanics and quantum fields , 1995 .

[7]  Shor,et al.  Good quantum error-correcting codes exist. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[8]  David Deutsch,et al.  Stabilisation of Quantum Computations by Symmetrisation , 1996 .

[9]  J. Bernardo.  Reference Posterior Distributions for Bayesian Inference , 1979 .

[10]  P. Slater.  Applications of quantum and classical Fisher information to two-level complex and quaternionic and three-level complex systems , 1996 .

[11]  J. Aitchison.  Goodness of prediction fit , 1975 .

[12]  Jozsa,et al.  General fidelity limit for quantum channels. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[13]  Sergio Verdú,et al.  Fifty Years of Shannon Theory , 1998, IEEE Trans. Inf. Theory.

[14]  David Deutsch,et al.  Stabilization of Quantum Computations by Symmetrization , 1997, SIAM J. Comput..

[15]  Cleve,et al.  Schumacher's quantum data compression as a quantum computation. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[16]  R. E. Krichevskii.  Universal Compression and Retrieval , 1994 .

[17]  S. G. Mohanty,et al.  Lattice Path Counting and Applications. , 1980 .

[18]  Andrew R. Barron,et al.  Information-theoretic asymptotics of Bayes methods , 1990, IEEE Trans. Inf. Theory.

[19]  Hiroshi Nagaoka,et al.  Quantum Fisher metric and estimation for pure state models , 1995 .

[20]  I. Good,et al.  Utility of a Distribution , 1968, Nature.

[21]  Fivel.  How interference effects in mixtures determine the rules of quantum mechanics. , 1994, Physical Review A. Atomic, Molecular, and Optical Physics.

[22]  D. Petz,et al.  Geometries of quantum states , 1996 .

[23]  K.R.W. Jones.  Principles of quantum inference , 1991 .

[24]  Rafail Krichevsky.  Universal Compression and Retrieval , 1994 .

[25]  A. Wehrl.  General properties of entropy , 1978 .

[26]  Massar,et al.  Optimal extraction of information from finite quantum ensembles. , 1995, Physical review letters.

[27]  Dénes Petz,et al.  The Bogoliubov inner product in quantum statistics , 1993 .

[28]  Andrew R. Barron,et al.  Jeffreys' prior yields the asymptotic minimax redundancy , 1994, Proceedings of 1994 Workshop on Information Theory and Statistics.

[29]  Schumacher,et al.  Quantum coding. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[30]  Stefano De Leo,et al.  Odd Dimensional Translation between Complex and Quaternionic Quantum Mechanics , 1996 .

[31]  Hoi-Kwong Lo.  Quantum coding theorem for mixed states , 1995 .

[32]  A. Srivastav,et al.  A characterization of the classical states of the quantum harmonic oscillator by means of de Finetti's theorem , 1989 .

[33]  S. Szarek,et al.  Volumes of restricted Minkowski sums and the free analogue of the entropy power inequality , 1995, math/9510203.

[34]  L. Wasserman,et al.  The Selection of Prior Distributions by Formal Rules , 1996 .

[35]  Lee D. Davisson,et al.  Universal noiseless coding , 1973, IEEE Trans. Inf. Theory.

[36]  D. Petz.  Geometry of canonical correlation on the state space of a quantum system , 1994 .

[37]  Bertrand Clarke,et al.  Implications of Reference Priors for Prior Information and for Sample Size , 1996 .

[38]  Feng Qi,et al.  Generalized weighted mean values with two parameters† , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[39]  Alberto Leon-Garcia,et al.  A source matching approach to finding minimax codes , 1980, IEEE Trans. Inf. Theory.

[40]  Mizan Rahman,et al.  Basic Hypergeometric Series , 1990 .

[41]  B. Dwork.  Generalized Hypergeometric Functions , 1990 .

[42]  Jorma Rissanen,et al.  Fisher information and stochastic complexity , 1996, IEEE Trans. Inf. Theory.

[43]  N. Vilenkin,et al.  Representation of Lie groups and special functions , 1991 .

[44]  D. Petz,et al.  Quantum Entropy and Its Use , 1993 .

[45]  A. Barron,et al.  Jeffreys' prior is asymptotically least favorable under entropy risk , 1994 .

[46]  Werner,et al.  Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. , 1989, Physical review. A, General physics.

[47]  R. A. Hutchinson,et al.  Image coding using an analysis-synthesis technique , 1990 .

[48]  David P. DiVincenzo,et al.  Quantum information and computation , 2000, Nature.

[49]  R. Schumann.  Quantum Information Theory , 2000, quant-ph/0010060.

[50]  Jorma Rissanen,et al.  Stochastic Complexity in Statistical Inquiry , 1989, World Scientific Series in Computer Science.

[51]  N. N. Chent︠s︡ov.  Statistical decision rules and optimal inference , 1982 .

[52]  Dénes Petz,et al.  On the Riemannian metric of α-entropies of density matrices , 1996 .

[53]  Shigeichi Hirasawa,et al.  A class of distortionless codes designed by Bayes decision theory , 1991, IEEE Trans. Inf. Theory.

[54]  N. Čencov.  Statistical Decision Rules and Optimal Inference , 2000 .

[55]  Dominic J. A. Welsh,et al.  Codes and cryptography , 1988 .

[56]  Karl Svozil.  Quantum algorithmic information theory , 1995 .

[57]  H. Dishkant,et al.  Logic of Quantum Mechanics , 1976 .

[58]  B. Frieden,et al.  Lagrangians of physics and the game of Fisher-information transfer. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[59]  Milburn,et al.  Dynamics of statistical distance: Quantum limits for two-level clocks. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[60]  Paul B. Slater,et al.  LETTER TO THE EDITOR: Quantum Fisher - Bures information of two-level systems and a three-level extension , 1996 .

[61]  N. Mermin.  Quantum theory: Concepts and methods , 1997 .

[62]  Paul B. Slater,et al.  LETTER TO THE EDITOR: The quantum Jeffreys' prior/Bures metric volume element for squeezed thermal states and a universal coding conjecture , 1996 .

[63]  Benjamin Schumacher,et al.  A new proof of the quantum noiseless coding theorem , 1994 .