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, $P_A(z)=P_{UAU^\dagger}(z)$. This is a consequence of the fact that the integration measure ${\rm d} \mu(\psi)$ is unitarily invariant.
3. For any normal matrix $A$, such that $AA^\dagger=A^\dagger A$, its shadow covers the numerical range $W(A)$ with the probability corresponding to a projection of a regular $N$–simplex of classical states ${\cal C}_N$ (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 $\Omega_N={\mathbb{C}}P^{N-1}$ 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, $S(\ket{z} \otimes \ket{y})=\ket{y} \otimes \ket{z}$. Thus $\bra{x} \left( A \otimes B \right) \ket{x} = \bra{x} S^{\dagger} \left( B \otimes A \right) S \ket{x}$, and since $S$ is unitary it does not influence the numerical shadow induced by the unitarily invariant Fubini-Study measure on complex projective space.