Separability criteria with angular and Hilbert space averages

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• 作者：Kazuo Fujikawa^a ; ^b ; ^{fujikawa@phys.s.u-tokyo.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; C.H. Oh^b ; ^c ; Koichiro Umetsu^d ; Sixia Yu^c
• 关键词：Separability ; Entanglement ; Two-particle correlation
• 刊名：Annals of Physics
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：368
• 期：Complete
• 页码：248-257
• 全文大小：418 K

文摘

The practically useful criteria of separable states 744756d2278b6dd4d614b408b" title="Click to view the MathML source">ρ=∑_kw_kρ_k$\rho ={\sum }_k{w}_k{\rho }_k$ in d=2×2$d=2\times 2$ are discussed. The equality $G(a,b)=4[〈\psi |P\left(a\right)\otimes P\left(b\right)|\psi 〉-〈\psi |P\left(a\right)\otimes 1|\psi 〉〈\psi |1\otimes P\left(b\right)|\psi 〉]=0$ for any two projection operators $P\left(a\right)$ and $P\left(b\right)$ provides a necessary and sufficient separability criterion in the case of a separable pure state ρ=|ψ〉〈ψ|$\rho =|\psi 〉〈\psi |$. We propose the separability criteria of mixed states, which are given by $\mathrm{Tr}\rho \{a\cdot \sigma \otimes b\cdot \sigma \}=(1/3)Ccos\phi$ for two spin 74cba438634e" title="Click to view the MathML source">1/2$1/2$ systems and $4\mathrm{Tr}\rho \{P\left(a\right)\otimes P\left(b\right)\}=1+(1/2)Ccos2\phi$ for two photon systems, respectively, after taking a geometrical angular average of 74cff43d63bad27fb582d">$a$ and $b$ with fixed $cos\phi =a\cdot b$. Here −1≤C≤1$-1\le C\le 1$, and the difference in the numerical coefficients 74cba438634e" title="Click to view the MathML source">1/2$1/2$ and 7288ebaa" title="Click to view the MathML source">1/3$1/3$ arises from the different rotational properties of the spinor and the transverse photon. If one instead takes an average over the states in the 72e36209c9f46cf" title="Click to view the MathML source">d=2$d=2$ Hilbert space, the criterion for two photon systems is replaced by $4\mathrm{Tr}\rho \{P\left(a\right)\otimes P\left(b\right)\}=1+(1/3)Ccos2\phi$. 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.