On the pseudo-manifold of quantum states

There are various statements in the physics literature about stratification of quantum states, for example into orbits a unitary group, and generalized differentiable structures on it. Our aim is to clarify make precise some these statements. For A an arbitrary finite-dimensional C*-algebra U(A) group elements A, we observe that partition state space S(A) not decomposition orbit types (its pieces manifolds without boundary), while there natural Whitney matrices fixed rank. latter, when full matrix algebra, give explicit description pseudo-manifold structure (the conical neighborhood around any point). We then comments infinite-dimensional case.