GHZ numerical shadow
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 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 $\Omega=\{\ket{\psi} \in \mathbb{C}^{2^d}: \ket{\psi} = \frac{1}{\sqrt{2}} \bigotimes_{i=1}^d U_i \left( \ket{0}^{\otimes d} + \ket{1}^{\otimes d} \right)\},$ where $U_i \in SU(2)$
Example
GHZ entangled numerical shadow of a unitary matrix [1] $U=\text{diag}\left( 1,e^{\frac{2 \ii \pi }{3}}, e^{\frac{2 \ii \pi }{3}}, e^{-\frac{2 \ii \pi }{3}}, e^{\frac{2 \ii \pi }{3}}, e^{-\frac{2 \ii \pi }{3}}, e^{-\frac{2 \ii \pi }{3}}, 1 \right)$
References
- [1]Z. Puchała, J. A. Miszczak, P. Gawron, C. F. Dunkl, J. A. Holbrook, and K. Życzkowski, “Restricted numerical shadow and geometry of quantum entanglement,” Journal of Physics A: Mathematical and Theoretical, vol. 45, no. 41, p. 415309, 2012, [Online]. Available at: https://d1wqtxts1xzle7.cloudfront.net/40538136/Restricted_numerical_shadow_and_geometry20151201-28343-1tqkl28.pdf?1448978875=&response-content-disposition=inline%3B+filename%3DRestricted_numerical_shadow_and_the_geom.pdf&Expires=1592565216&Signature=Hejde5pW2ID8kfVbbr4wAHu9SsM221OAQUiQ21j6SW-rMFjUCgWyXtlxji-gowYWowFcbjFHF4XMCc3ieuGOozuW9PiuPtPvd W7pK6tfWiTywFcNEbu3XfuC5fbYdb6A zL5kIlS6yKX5E4Bldoe-V424Mbk6JehrCaJ7-HEL8kYH21aZt DAI7RX4BEF cRtjTVYf8I0PcJAZMIn iCum0D0sI1MsMCYpnUHHe0J-WpGnDGo509mszZlIZYfAzoQfdVPpCRVhc6WUvkGnI5Eeyl6 NjJ4mEjvCAOXETEcNtl1ktGKRtMK 5pDe96SsPTTg3JQBg8YtgqvZlQJKlQ__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA.