Numerical Shadow

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numerical-shadow:generalizations:restricted-numerical-shadow:ghz-numerical-shadow

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numerical-shadow:generalizations:restricted-numerical-shadow:ghz-numerical-shadow [2013/08/03 21:03]
lpawela
numerical-shadow:generalizations:restricted-numerical-shadow:ghz-numerical-shadow [2018/10/08 08:58] (current)
plewandowska
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-GHZ entangled numerical shadow of a matrix $A$ is defined as a probability ditribution $P_A(z)$ on the complex plane, supported on the [[numerical-shadow:​generalizations:​restricted-numerical-shadow:​entangled-numerical-shadow|maximally entangled numerical range]] $W^\mathrm{ent}(A)$.+GHZ entangled numerical shadow of a matrix $A$ of dimension $d$ is defined as a probability ditribution $P_A(z)$ on the complex plane, supported on the [[numerical-shadow:​generalizations:​restricted-numerical-shadow:​entangled-numerical-shadow|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),​ P_A(z) := \int_{\Omega} {\rm d} \mu(\psi) \delta\Bigl( z-\langle \psi|A|\psi\rangle\Bigr),​
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 where $\mu(\psi)$ denotes the unique unitarily invariant (Fubini-Study) measure on the set where $\mu(\psi)$ denotes the unique unitarily invariant (Fubini-Study) measure on the set
 $$ $$
-\Omega=\{\ket{\psi} \in \mathbb{C}^N: \ket{\psi} = \frac{1}{\sqrt{2}} \bigotimes_{i=1}^U_i \left( \ket{0}^{\otimes ​N} + \ket{1}^{\otimes ​N} \right), \;​\braket{\psi}{\psi}=1\},​+\Omega=\{\ket{\psi} \in \mathbb{C}^d: \ket{\psi} = \frac{1}{\sqrt{2}} \bigotimes_{i=1}^U_i \left( \ket{0}^{\otimes ​d} + \ket{1}^{\otimes ​d} \right), \;​\braket{\psi}{\psi}=1\},​
 $$ $$
 where $U_i \in SU(2)$ where $U_i \in SU(2)$
numerical-shadow/generalizations/restricted-numerical-shadow/ghz-numerical-shadow.txt · Last modified: 2018/10/08 08:58 by plewandowska