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Product numerical shadow


Product numerical shadow of a matrix $A$ of dimension $d$ is defined as a probability distribution $P_A(z)$ on the complex plane, supported on the product numerical range $W^\otimes(A)$. $P_A(z) := \int_{\Omega} {\rm d} \mu(\psi) \delta\Bigl( z- \bra{\psi} A \ket{\psi} \Bigr),$ where $\mu(\psi)$ denotes the unique local unitarily invariant (Fubini-Study) measure on the set $\Omega = \{ \ket{\psi} \in \mathbb{C} ^ d: \ket{\psi} = \bigotimes_{i=1}^d \ket{\phi_i}, \text{ for } i=1,\ldots,d\ \braket{\phi_i}{\phi_i}=1 \text{ and } \ket{\phi_i}\in\mathbb{C}^2 \}$