Real numerical shadow and generalized B-splines

Restricted numerical shadow $P_A^X(z)$ of an operator $A$ of order $N$ is a probability distribution supported on the numerical range $W_X(A)$ restricted to a certain subset $X$ of the set of all pure states - normalized, one-dimensional vectors in ${\mathbb{C}}^N$. Its value at point $z \in {\mathbb C}$equalstotheprobabilitythattheinnerproduct$< u |A| u >$isequalto$z$,where$u$standsforarandomcomplexvectorfromtheset$X$ 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, $P_A^{\mathbb{R}}(x)$, 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, ${\mathcal{H}}_A\otimes {\mathcal{H}}_B$, we analyze the shadow restricted to the set of maximally entangled states and derive distributions for operators of order N=4.