局部实在论Peres判据的非广延熵推广
Generalization of the Peres criterion for local realism through nonextensive entropy
摘要：
已知一个自旋为1/2的二分系统，其概率$\frac{1+3x}{4}$处于Einstein-Podolsky-Rosen纠缠态$|Ψ^-$ $> \equiv \frac{1}{\sqrt 2}(|$ $\uparrow>_A|$ $\downarrow>_B$ $-|$ $\downarrow>_A|$ PDMATHAMT和$\frac{3(1-x)}{4}$是正交的，当且仅当$x<1/3$（Peres判据）时，该系统允许局域真实描述。我们在这里考虑一个更一般的情况下，在纠缠态$|Φ^{\pm}$ $> \equiv \frac{1}{\sqrt 2}(|$ $\uparrow>_A|$ $\uparrow>_B \pm |$ $\downarrow>_A|$ $\downarrow>_B)$和$|Ψ^{\pm}$ $> \equiv \frac{1}{\sqrt 2}(|$ $\uparrow>_A|$ $\downarrow>_B \pm |$ $\downarrow>_A|$ $\uparrow>_B)$（贝尔基）的概率分别由$\frac{1-x}{4}$，$\frac{1-y}{4}$，$\frac{1-z}{4}$和$\frac{1+x+y+z}{4}$给出。在Abe和Rajagopal之后，我们使用非广延熵形式$S_q \equiv \frac{1- Tr ρ^q}{q-1} (q \in \cal{R}; $ $S_1$ $= -$ $Tr$ $ ρ\ln ρ)$，它使得玻尔兹曼-吉布斯统计力学能够得到当前的推广，并确定了$(x,y,z)$空间中局部实在论是允许的整个区域。例如，在EPR状态附近，经典实在论是可能的，当且仅当$x+y+z<1$，当$x=y=z$恢复佩雷斯的标准。在贝尔基的其他三个状态附近，情况是相同的。一个类似临界现象的场景出现了。这些结果说明了这种新的非广延量子信息程序的计算能力。
英文摘要：
A bipartite spin-1/2 system having the probabilities $\frac{1+3x}{4}$ of being in the Einstein-Podolsky-Rosen entangled state $|Ψ^-$$> \equiv \frac{1}{\sqrt 2}(|$$\uparrow>_A|$$\downarrow>_B$$-|$$\downarrow>_A|$$\uparrow>_B)$ and $\frac{3(1-x)}{4}$ of being orthogonal, is known to admit a local realistic description if and only if $x<1/3$ (Peres criterion). We consider here a more general case where the probabilities of being in the entangled states $|Φ^{\pm}$$> \equiv \frac{1}{\sqrt 2}(|$$\uparrow>_A|$$\uparrow>_B \pm |$$\downarrow>_A|$$\downarrow>_B)$ and $|Ψ^{\pm}$$> \equiv \frac{1}{\sqrt 2}(|$$\uparrow>_A|$$\downarrow>_B \pm |$$\downarrow>_A|$$\uparrow>_B)$ (Bell basis) are given respectively by $\frac{1-x}{4}$, $\frac{1-y}{4}$, $\frac{1-z}{4}$ and $\frac{1+x+y+z}{4}$. Following Abe and Rajagopal, we use the nonextensive entropic form $S_q \equiv \frac{1- Tr ρ^q}{q-1} (q \in \cal{R}; $$S_1$$= -$ $Tr$ $ ρ\ln ρ)$ which has enabled a current generalization of Boltzmann-Gibbs statistical mechanics, and determine the entire region in the $(x,y,z)$ space where local realism is admissible. For instance, in the vicinity of the EPR state, classical realism is possible if and only if $x+y+z<1$, which recovers Peres' criterion when $x=y=z$. In the vicinity of the other three states of the Bell basis, the situation is identical. A critical-phenomenon-like scenario emerges. These results illustrate the computational power of this new nonextensive-quantum-information procedure.
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