Real numerical shadow and generalized B-splines

Restricted numerical shadow PAX(z) of an operator A of order N is a probability distribution supported on the numerical range WX(A) restricted to a certain subset X of the set of all pure states - normalized, one-dimensional vectors in CN. Its value at point z∈C equals the probability that the inner product 〈u|A|u〉 is equal to z, where u stands for a random complex vector from the set X distributed according to the natural measure on this set, induced by the unitarily invariant Fubini-Study measure. For a Hermitian operator A of order N we derive an explicit formula for its shadow restricted to real states, PAR(x), show relation of this density to the Dirichlet distribution and demonstrate that it forms a generalization of the B-spline. Furthermore, for operators acting on a space with tensor product structure, HA⊗HB, we analyze the shadow restricted to the set of maximally entangled states and derive distributions for operators of order N=4.