  The practically useful criteria of separable states
$\rho=\sum_{k}w_{k}\rho_{k}$ in $d=2\times2$ are discussed. The equality
$G({\bf a},{\bf b})= 4[\langle \psi|P({\bf a})\otimes P({\bf
b})|\psi\rangle-\langle \psi|P({\bf a})\otimes{\bf 1}|\psi\rangle\langle
\psi|{\bf 1}\otimes P({\bf b})|\psi\rangle]=0$ for any two projection operators
$P({\bf a})$ and $P({\bf b})$ provides a necessary and sufficient separability
criterion in the case of a separable pure state
$\rho=|\psi\rangle\langle\psi|$. We propose the separability criteria of mixed
states, which are given by ${\rm Tr}\rho\{{\bf a}\cdot {\bf \sigma}\otimes {\bf
b}\cdot {\bf \sigma}\}=(1/3)C\cos\varphi$ for two spin $1/2$ systems and $4{\rm
Tr}\rho \{P({\bf a})\otimes P({\bf b})\}=1+(1/2)C\cos2\varphi$ for two photon
systems, respectively, after taking a geometrical angular average of ${\bf a}$
and ${\bf b}$ with fixed $\cos\varphi={\bf a}\cdot{\bf b}$. Here $-1\leq C\leq
1$, and the difference in the numerical coefficients $1/2$ and $1/3$ arises
from the different rotational properties of the spinor and the transverse
photon. If one instead takes an average over the states in the $d=2$ Hilbert
space, the criterion for two photon systems is replaced by $4{\rm Tr}\rho
\{P({\bf a})\otimes P({\bf b})\}=1+(1/3)C\cos2\varphi$.
  Those separability criteria are shown to be very efficient using the existing
experimental data of Aspect et al. in 1981 and Sakai et al. in 2006. When the
Werner state is applied to two photon systems, it is shown that the Hilbert
space average can judge its inseparability but not the geometrical angular
average.
