Properties of numerical shadow

1. By construction the distribution $P_A(z)$ is supported on the numerical range of $W(A)$ and it is normalized,

   $${\int }_{W(A)}{P}_A(z)d^{2}z=1.$$
2. The (numerical) shadow is unitarily invariant, $P_A(z)=P_{UA{U}^{†}}(z)$. This is a consequence of the fact that the integration measure $\mathrm{d}\mu (\psi )$ is unitarily invariant.
3. For any normal matrix $A$, such that $A{A}^{†}=A^{†}A$, its shadow covers the numerical range $W(A)$ with the probability corresponding to a projection of a regular $N$–simplex of classical states ${\mathcal{C}}_N$ (embedded in $R^{N-1}$) onto a plane.
4. For a non–normal matrix $A$, its shadow covers the numerical range $W(A)$ with the probability corresponding to an orthogonal projection of the complex projective manifold ${\mathrm{\Omega }}_N=\mathbb{C}{P}^{N-1}$ onto a plane.

5. For any two matrices $A$ and $B$, the shadow of their tensor product does not depend on the order, $P_{A\otimes B}(z)=P_{B\otimes A}(z).$

   To show this property, define a unitary swap operator $S$ which acts on a composite Hilbert space and interchanges the order in the tensor product, $S(|z⟩\otimes |y⟩)=|y⟩\otimes |z⟩$. Thus $⟨x|(A\otimes B)|x⟩=⟨x|S^{†}(B\otimes A)S|x⟩$, and since $S$ is unitary it does not influence the numerical shadow induced by the unitarily invariant Fubini-Study measure on complex projective space.