Separability criteria with angular and Hilbert space averages
  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. 
Kazuo Fujikawa, C.H. Oh, Koichiro Umetsu, Sixia Yu
quant-ph