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 $ u=2$ are discussed, where Haldanes sphere picture is generalized to a Grassmannian picture. We also extend Wehrls definition of entropy from Glauber to Grassmannian CS and state a conjecture on the entropy lower bound.
