PDF (double-sided) - Physics Department, UCSB - University of ...

This restriction is called the Bell inequality. The beauty of this inequality is
that its derivation does not assume anything about the used measurement axes
a, a ′ , b and b ′ or even about the fractions n i , i.e. about the distribution of pairs
that the source produces. This allows for a lot of freedom in the implementation
and makes the inequality very robust against imperfections.
It turns out that there are choices for the possible measurement axes a, a ′ ,
b and b ′
for which quantum mechanics predicts a value that violates the Bell
inequality for certain pair states. To give an example, the source can be taken to
prepare particle pairs in the Singlet state | ψ 〉 =
| 01 〉−| 10 〉
√
2
and all measurements
can be confined to axes to the X/Z-plane. The measurement axes a, a ′ , b, and
b ′ can then be specified by the angles they form with the Z-axis, α, α ′ , β, and
β ′ . Quantum mechanics predicts the probability P | 1 〉 of a positive outcome of a
measurement around an axis x of a single particle in state | ψ 〉 in the Z-basis | 0 〉
and | 1 〉 as:
P | 1 〉,ψ (θ) = |〈 θ | ψ 〉| 2 =
∣
(cos θ 2 〈 0 | + sin θ )
2 〈 1 | | ψ 〉
∣
2
(10.8)
The probability P | 0 〉 of a negative outcome can be seen as the probability of a
positive outcome if the measurement had been in the opposite direction, i.e.:
P | 0 〉,ψ (θ) = P | 1 〉,ψ (θ + 180 ◦ ) (10.9)
231