  The maximum observable correlation between the two components of a bipartite
quantum system is a property of the joint density operator, and is achieved by
making particular measurements on the respective components. For pure states it
corresponds to making measurements diagonal in a corresponding Schmidt basis.
More generally, it is shown that the maximum correlation may be characterised
in terms of a `correlation basis' for the joint density operator, which defines
the corresponding (nondegenerate) optimal measurements. The maximum coincidence
rate for spin measurements on two-qubit systems is determined to be (1+s)/2,
where s is the spectral norm of the spin correlation matrix, and upper bounds
are obtained for n-valued measurements on general bipartite systems. It is
shown that the maximum coincidence rate is never greater than the computable
cross norm measure of entanglement, and a much tighter upper bound is
conjectured. Connections with optimal state discrimination and entanglement
bounds are briefly discussed.
