Numerical Shadow

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

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numerical-shadow:generalizations:restricted-numerical-shadow:entangled-numerical-shadow [2013/11/08 13:43]
lpawela
numerical-shadow:generalizations:restricted-numerical-shadow:entangled-numerical-shadow [2018/10/08 08:57] (current)
plewandowska [Definition]
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 ===== Definition ===== ===== 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 [[numerical-range:​generalizations:​restricted-numerical-range:​maximally-entangled-numerical-range|maximally entangled numerical range]] $W^\mathrm{ent}(A)$.+Maximally entangled numerical shadow of a matrix $A$ of size $d=d_1\times ​d_2$ is defined as a probability distribution $P_A(z)$ on the complex plane, supported on the [[numerical-range:​generalizations:​restricted-numerical-range:​maximally-entangled-numerical-range|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_1\times ​N_2}: +\Omega=\{\ket{\psi} \in \mathbb{C}^{d_1\times ​d_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} \},+\ket{\psi}=\frac{1}{\sqrt{d_\min}} (U_1\otimes U_2)\sum_{i=1}^{d_\min} \ket{\psi_i^1}\otimes \ket{\psi_i^2} \},
 $$ $$
 where where
-  * $N_\min={\min(N_1,N_2)}$,  +  * $d_\min={\min(d_1,d_2)}$,  
-  * $\ket{\psi_i^1}$,​ $\ket{\psi_i^2}$ form orthonormal bases in $\mathbb{C}^{N_1}$ and $\mathbb{C}^{N_2}$ respectively,​ +  * $\ket{\psi_i^1}$,​ $\ket{\psi_i^2}$ form orthonormal bases in $\mathbb{C}^{d_1}$ and $\mathbb{C}^{d_2}$ respectively,​ 
-  * $U_1\in SU(N_1)$ and $U_2\in SU(N_2)$.+  * $U_1\in SU(d_1)$ and $U_2\in SU(d_2)$.
  
 ===== Examples ===== ===== Examples =====
  
   * [[numerical-shadow:​examples:​4x4#​entangled_numerical_shadow|Entangled shadow of 4x4 non-normal matrices]]   * [[numerical-shadow:​examples:​4x4#​entangled_numerical_shadow|Entangled shadow of 4x4 non-normal matrices]]
numerical-shadow/generalizations/restricted-numerical-shadow/entangled-numerical-shadow.txt · Last modified: 2018/10/08 08:57 by plewandowska