Asymptotic redundancies for universal quantum coding

Clarke and Barren have recently shown that the Jeffreys' invariant prior of Bayesian theory yields the common asymptotic (minimax and maximin) redunda ncy of universal data compression in a parametric setting. We seek a possib le analog of this result for the two-level quantum systems, We restrict our considerations to prior probability distributions belonging to a certain o ne-parameter family, q(u), - infinity < 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 der ived by naively applying the (nonquantum) counterparts of Clarke and Barren , and find certain common features, Our results are based on formulas we ob tain for the eigenvalues and eigenvectors of 2(n) x 2(n) (Bayesian density) matrices, zeta(n)(u), These matrices are the weighted averages (with respe ct to q(u)) of all possible tensor products of n identical 2 x 2 density ma trices, representing the two-level quantum systems, We propose a form of un iversal 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 zeta(n)(u).