  We analyze mathematical and physical properties of a previously introduced
[J. Phys. A47, 115302 (2014)] family of $U(4)$ coherent states (CS). They
constitute a matrix version of standard spin $U(2)$ CS when we add an extra
(pseudospin) dichotomous degree of freedom: layer, sublattice, two-well,
nucleon, etc. Applications to bilayer quantum Hall systems at fractions of
filling factor $\nu=2$ are discussed, where Haldane's sphere picture is
generalized to a Grassmannian picture. We also extend Wehrl's definition of
entropy from Glauber to Grassmannian CS and state a conjecture on the entropy
lower bound.
