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).

Internal migration tables can be employed to ascertain functional migration regions-collections of geographic units with relatively large intraregional flows and relatively small interregional movements. A two-stage algorithm that has been utilized to study tables for various nations is applied here to a 21 × 21 French matrix. In the first step, the square table is adjusted to possess uniform marginal (row and column) sums. The doubly standardized table obtained can be regarded as an asymmetric similarity matrix. It serves, in the second stage, as input to a hierarchical clustering procedure, based on the concep of the strong components of a directed graph. Région Parisienne and Centre are shown to have the broadest migration bases. Well-defined southern and northwestern regions are found.

Data on the international movement of petroleum are compiled by the United Nations. Two graphical devices are employed to facilitate an understanding of the patterns in the 1970 data. They both require the estimation of an origin-destination independence model of flows. Residuals from the model are examined, and results interpreted.