# Numerical Shadow

The web resource on numerical range and numerical shadow

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numerical-shadow:generalizations:restricted-numerical-shadow:w-numerical-shadow

# 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 $$\Omega=\{\ket{\psi} \in \mathbb{C}^{2^d}: \ket{\psi} = \frac{1}{\sqrt{d}} \bigotimes_{i=1}^d U_i \left( \ket{10\ldots 0} + \ket{01 \ldots 0} + \ldots + \ket{00 \ldots 1} \right)\},$$ where $U_i \in SU(2)$.

## Example

W entanglement numerical shadow of a unitary matrix [1] $$U=\text{diag}\left( 1,e^{\frac{2 \ii \pi }{3}}, e^{\frac{2 \ii \pi }{3}}, e^{-\frac{2 \ii \pi }{3}}, e^{\frac{2 \ii \pi }{3}}, e^{-\frac{2 \ii \pi }{3}}, e^{-\frac{2 \ii \pi }{3}}, 1 \right)$$

1. Z. Puchała, J.A. Miszczak, P. Gawron, C.F. Dunkl, J.A. Holbrook, K. Życzkowski, 2012. Restricted numerical shadow and geometry of quantum entanglement. Journal of Physics A: Mathematical and Theoretical, 45, pp.415309.
numerical-shadow/generalizations/restricted-numerical-shadow/w-numerical-shadow.txt · Last modified: 2019/10/01 12:44 by rkukulski