Numerical Shadow

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W numerical shadow

W entangled 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 maximally entangled numerical range $W^\mathrm{ent}(A)$. $$ P_A(z) := \int_{\Omega} {\rm d} \mu(\psi) \delta\Bigl( z-\langle \psi|A|\psi\rangle\Bigr), $$ where $\mu(\psi)$ denotes the unique unitarily invariant (Fubini-Study) measure on the set $$ \Omega=\{\ket{\psi} \in \mathbb{C}^d: \ket{\psi} = \frac{1}{\sqrt{d}} \bigotimes_{i=1}^d U_i \left( \ket{10\ldots 0} + \ket{01 \ldots 0} + \ldots + \ket{00 \ldots 1} \right), \;\braket{\psi}{\psi}=1\}, $$ where $U_i \in SU(2)$.

numerical-shadow/generalizations/restricted-numerical-shadow/w-numerical-shadow.txt · Last modified: 2018/10/08 08:59 by plewandowska