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arXiv 1603.01792quant-ph
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角平均和Hilbert空间平均的可分性准则

Separability criteria with angular and Hilbert space averages

Kazuo Fujikawa, C. H. Oh, Koichiro Umetsu, Sixia Yu

备注：
14 pages, 3 figures
发表信息：
Annals of Physics 368 (2016) 248
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摘要：

讨论了$d=2\times2$中可分态$ρ=\sum_{k}w_{k}ρ_{k}$的实用判据。在可分纯态$ρ=|ψ\rangle\langleψ|$的情况下，任何两个投影算符$P({\bf a})$和$P({\bf b})$的等式$G({\bf a},{\bf b})= 4[\langle ψ|P({\bf a})\otimes P({\bf b})|ψ\rangle-\langle ψ|P({\bf a})\otimes{\bf 1}|ψ\rangle\langle ψ|{\bf 1}\otimes P({\bf b})|ψ\rangle]=0$提供了一个必要且充分的可分性判据.在对固定$\cosφ={\bf a}\cdot{\bf b}$的${\bf a}$和${\bf b}$进行几何角平均后，我们提出了混合态的可分性判据，分别由双自旋$1/2$系统的${\rm Tr}ρ\{{\bf a}\cdot {\bf σ}\otimes {\bf b}\cdot {\bf σ}\}=(1/3)C\cosφ$和双光子系统的$4{\rm Tr}ρ\{P({\bf a})\otimes P({\bf b})\}=1+(1/2)C\cos2φ$给出.这里是$-1\leq C\leq 1$，数值系数$1/2$和$1/3$的差异是由旋量和横向光子的不同旋转性质引起的。如果在$d=2$希尔伯特空间中对状态取平均值，则双光子系统的判据由$4{\rm Tr}ρ\{P({\bf a})\otimes P({\bf b})\}=1+(1/3)C\cos2φ$代替。这些可分性准则被证明是非常有效的，使用现有的实验数据方面等。在1981年和Sakai等。在2006年。当Werner态应用于双光子系统时，证明了Hilbert空间平均可以判断其不可分性，而几何角平均不能。

英文摘要：

The practically useful criteria of separable states $ρ=\sum_{k}w_{k}ρ_{k}$ in $d=2\times2$ are discussed. The equality $G({\bf a},{\bf b})= 4[\langle ψ|P({\bf a})\otimes P({\bf b})|ψ\rangle-\langle ψ|P({\bf a})\otimes{\bf 1}|ψ\rangle\langle ψ|{\bf 1}\otimes P({\bf b})|ψ\rangle]=0$ 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 $ρ=|ψ\rangle\langleψ|$. We propose the separability criteria of mixed states, which are given by ${\rm Tr}ρ\{{\bf a}\cdot {\bf σ}\otimes {\bf b}\cdot {\bf σ}\}=(1/3)C\cosφ$ for two spin $1/2$ systems and $4{\rm Tr}ρ\{P({\bf a})\otimes P({\bf b})\}=1+(1/2)C\cos2φ$ for two photon systems, respectively, after taking a geometrical angular average of ${\bf a}$ and ${\bf b}$ with fixed $\cosφ={\bf a}\cdot{\bf b}$. 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 $4{\rm Tr}ρ\{P({\bf a})\otimes P({\bf b})\}=1+(1/3)C\cos2φ$. 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.

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