Asymptotic Redundancies for Universal Quantum Coding

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 $\ze_n(u)$.