Numerical Shadow

The web resource on numerical range and numerical shadow

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numerical-shadow [2018/05/22 14:43]
plewandowska [Other names]
numerical-shadow [2018/10/08 08:50] (current)
plewandowska [Definition]
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 ===== Definition ===== ===== Definition =====
  
-For any square matrix $A$ of size $N$, one defines its //numerical shadow// as a probability distribution $P_A(z)$ on the complex plane, supported in the numerical range $W(A)$, ​+For any square matrix $A$ of dimension ​$d$, one defines its //numerical shadow// as a probability distribution $P_A(z)$ on the complex plane, supported in the numerical range $W(A)$, ​
 $$ $$
-P_A(z) := \int_{\Omega_N} {\rm d} \mu(\psi) \delta\Bigl( z-\langle \psi|A|\psi\rangle\Bigr) . +P_A(z) := \int_{\Omega_d} {\rm d} \mu(\psi) \delta\Bigl( z-\langle \psi|A|\psi\rangle\Bigr) . 
 $$ $$
-Here $\mu(\psi)$ denotes the unique unitarily invariant (Fubini-Study) measure on the set $\Omega_N$ of $N$-dimensional pure quantum states. In other words the shadow $P$ of matrix $A$ at a given point $z$ characterizes the likelihood that the expectation value of $A$ among a random pure state is equal to $z$.+Here $\mu(\psi)$ denotes the unique unitarily invariant (Fubini-Study) measure on the set $\Omega_d$ of $N$-dimensional pure quantum states. In other words the shadow $P$ of matrix $A$ at a given point $z$ characterizes the likelihood that the expectation value of $A$ among a random pure state is equal to $z$.
  
 ===== Other names ===== ===== Other names =====
numerical-shadow.1527000186.txt.gz · Last modified: 2018/05/22 14:43 by plewandowska