  The purpose of this article is to investigate the geometry of the set of
locally diagonalizable bipartite quantum states. We have the following new
results: the Hilbert-Schmidt volume of all locally diagonalizable states, and a
necessary and sufficient condition for local diagonalizability in the
qubit-qubit case. Besides, we partition the set of all locally diagonalizable
states as local unitary orbits (or coadjoint orbits) of diagonal forms. It is
well-known that the Riemannian volume of a coadjoint orbit for a regular point
in a specified Weyl chamber can be calculated by Harish-Chandra's volume
formula. By modifying Harish-Chandra's volume formula, we give, for the first
time, a specific formula for the Riemannian volume of a local unitary orbit of
a regular point in a specified Weyl chamber. Several open questions are
presented as well.
