Asymptotic Redundancies for Universal Quantum Coding

Abstract

Clarke and Barron have recently 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 analogue of this result for the two-level {\it quantum} systems. We restrict our considerations to prior probability distributions belonging to a certain one-parameter family, $q(u)$, $-\infty <u<1$. 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 classical counterparts of Clarke and Barron, and find certain common features. Our results are based on formulas we obtain for the eigenvalues and eigenvectors of $2^n\times 2^n$ (Bayesian density) matrices, ${\zeta }_n(u)$. These matrices are the weighted averages (with respect to $q(u)$) of all possible tensor products of $n$ identical $2\times 2$ density matrices, representing the two-level quantum systems. We propose a form of {\it 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 $\ze_n(u)$.

Cited by in corpus (14)

• Quantum universal variable-length source coding
• Universal coding for classical-quantum channel
• Hall Normalization Constants for the Bures Volumes of the n-State Quantum Systems
• Silver mean conjectures for 15-d volumes and 14-d hyperareas of the separable two-qubit systems
• Qubit-Qutrit Separability-Probability Ratios
• Comparative Noninformativities of Quantum Priors Based on Monotone Metrics
• Universal approximation of multi-copy states and universal quantum lossless data compression
• Ratios of maximal concurrence-parameterized separability functions, and generalized Peres-Horodecki conditions
• Increased Efficiency of Quantum State Estimation Using Non-Separable Measurements
• Quantum and Fisher Information from the Husimi and Related Distributions
• Quantum Statistical Thermodynamics of Two-Level Systems
• Quasirandom estimations of two-qubit operator-monotone-based separability probabilities
• Volume Elements of Monotone Metrics on the n x n Density Matrices as Densities-of-States for Thermodynamic Purposes. II
• The Brody-Hughston Fisher Information Metric