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纠缠态的局域交换性与贝尔不等式破坏和可分态的非破坏

Local commutativity versus Bell inequality violation for entangled states and versus non-violation for separable states

Michael Seevinck, Jos Uffink

备注：
V2: Most important change: we have greatly extended the discussion of the theoretical and experimental relevance of the results. v3: Published version
发表信息：
Phys. Rev. A 76, 042105 (2007)
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摘要：

通过引入一个与自旋可观测量夹角有关的定量的“交换度”，我们给出了两个量子比特情形下的两个紧密的定量权衡关系：首先，对于纠缠态，局域可观测量的交换度与违反贝尔不等式的最大量之间：如果两个局域夹角都从零增加到π/2（即，局部交换性的程度降低），贝尔不等式的最大违反增加。其次，对于可分离态，存在一个相反的权衡关系：如果两个局部角都接近π/2，则贝尔不等式中的相关性可获得的最大值减小，因此非破坏性增加。正如预期的那样，这些关系的极端的情况下，发现反交换本地观测值分别为2 \sqrt{2}和\sqrt{2}的边界举行的贝尔运营商的期望。权衡关系表明，非交换性给出了“一个比经典的结果”的纠缠态，而“一个小于经典的结果”得到的可分离状态。可分离状态的权衡关系的实验相关性在于，它提供了对两个量子比特纠缠的实验测试。其优点有两方面：与违反贝尔不等式相比，它是一个更强的标准，并且与纠缠见证相比，它需要对实验中实现的可观测量做出不太强的假设。

英文摘要：

By introducing a quantitative `degree of commutativity' in terms of the angle between spin-observables we present two tight quantitative trade-off relations in the case of two qubits: First, for entangled states, between the degree of commutativity of local observables and the maximal amount of violation of the Bell inequality: if both local angles increase from zero to π/2 (i.e., the degree of local commutativity decreases), the maximum violation of the Bell inequality increases. Secondly, a converse trade-off relation holds for separable states: if both local angles approach π/2 the maximal value obtainable for the correlations in the Bell inequality decreases and thus the non-violation increases. As expected, the extremes of these relations are found in the case of anti-commuting local observables where respectively the bounds of 2\sqrt{2} and \sqrt{2} hold for the expectation of the Bell operator. The trade-off relations show that non-commmutativity gives ``a more than classical result" for entangled states, whereas "a less than classical result" is obtained for separable states. The experimental relevance of the trade-off relation for separable states is that it provides an experimental test for two qubit entanglement. Its advantages are twofold: in comparison to violations of Bell inequalities it is a stronger criterion and in comparison to entanglement witnesses it needs to make less strong assumptions about the observables implemented in the experiment.

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