  Decomposition of any N-partite state (density operator) into clusters (that
do not overlap) is studied in detail with a view to learn as much as possible
about the correlations implied by the state. The Wootters-Mermin theorem,
stating that the totality of all strings of cluster events (projectors)
determines the state in any finite- or infinite-dimensional state space, is a
slightly sharpened and generalized form of the original results of Wootters and
Mermin. It is applied to tensor factorization of the state into states of
clusters (uncorrelated decomposition) and it is shown that a finest
uncorrelated decomposition always exists, and that its coarsenings and only
they are other possible uncorrelated cluster decompositions. Distant effects
witin homogeneous cluster states, which are, by definition, the tensor factors
in the finest uncorrelated decomposition, are discussed. The entire study is
viewed by the author as a possible further elaboration of Mermin's Ithaca
program.
