Numerical Shadow

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numerical-shadow:generalizations:restricted-numerical-shadow:product-numerical-shadow

Product numerical shadow

Definition

Product numerical shadow of a matrix $A$ 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 $$ \Omega = \{ \ket{\psi} \in \mathbb{C} ^ N: \ket{\psi} = \bigotimes_{i=1}^N \ket{\phi_i}, \text{ for } i=1,\ldots,N\ \braket{\phi_i}{\phi_i}=1 \text{ and } \ket{\phi_i}\in\mathbb{C}^2 \} $$

Examples

numerical-shadow/generalizations/restricted-numerical-shadow/product-numerical-shadow.txt · Last modified: 2013/11/08 14:53 by lpawela