Numerical Shadow

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

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numerical-shadow:generalizations:restricted-numerical-shadow:product-numerical-shadow [2013/11/08 14:53]
lpawela
numerical-shadow:generalizations:restricted-numerical-shadow:product-numerical-shadow [2018/10/08 08:56]
plewandowska [Definition]
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-Product numerical shadow of a matrix $A$ is defined as a probability distribution $P_A(z)$ on the complex plane, supported on the [[numerical-range:​generalizations:​restricted-numerical-range:​product-numerical-range|product numerical range]] $W^\otimes(A)$.+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 [[numerical-range:​generalizations:​restricted-numerical-range:​product-numerical-range|product numerical range]] $W^\otimes(A)$.
 $$ $$
 P_A(z) := \int_{\Omega} {\rm d} \mu(\psi) \delta\Bigl( z- \bra{\psi} A \ket{\psi} \Bigr), P_A(z) := \int_{\Omega} {\rm d} \mu(\psi) \delta\Bigl( z- \bra{\psi} A \ket{\psi} \Bigr),
Line 10: Line 10:
 $$ $$
 \Omega = \{ \Omega = \{
-\ket{\psi} \in \mathbb{C} ^ N:+\ket{\psi} \in \mathbb{C} ^ d:
 \ket{\psi} = \ket{\psi} =
-\bigotimes_{i=1}^N+\bigotimes_{i=1}^d
 \ket{\phi_i},​ \ket{\phi_i},​
-\text{ for } i=1,\ldots,N\ \braket{\phi_i}{\phi_i}=1 \text{ and } \ket{\phi_i}\in\mathbb{C}^2+\text{ for } i=1,\ldots,d\ \braket{\phi_i}{\phi_i}=1 \text{ and } \ket{\phi_i}\in\mathbb{C}^2
 \} \}
 $$ $$
numerical-shadow/generalizations/restricted-numerical-shadow/product-numerical-shadow.txt · Last modified: 2018/10/08 08:56 by plewandowska