Numerical Shadow

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numerical-shadow:generalizations:restricted-numerical-shadow:entangled-numerical-shadow

Maximally entangled numerical shadow

Definition

Maximally entangled numerical shadow of a matrix $A$ of size $N=N_1\times N_2$ 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 $$ \Omega=\{\ket{\psi} \in \mathbb{C}^{N_1\times N_2}: \ket{\psi}=\frac{1}{\sqrt{N_\min}} (U_1\otimes U_2)\sum_{i=1}^{N_\min} \ket{\psi_i^1}\otimes \ket{\psi_i^2} \}, $$ where

  • $N_\min={\min(N_1,N_2)}$,
  • $\ket{\psi_i^1}$, $\ket{\psi_i^2}$ form orthonormal bases in $\mathbb{C}^{N_1}$ and $\mathbb{C}^{N_2}$ respectively,
  • $U_1\in SU(N_1)$ and $U_2\in SU(N_2)$.

Examples

numerical-shadow/generalizations/restricted-numerical-shadow/entangled-numerical-shadow.txt · Last modified: 2013/11/08 13:43 by lpawela