Bell's inequality when efficiency < 1

[naima]

Hi PF

Somebody gave me this link
could you help me to understand why (9) has to be inserted in (7) when the efficiency \eta &lt; 1?

 

[naima]

I think that a part of the answer is here
but i do not understand eq (6):
&lt;ab&gt;_{coinc} = &lt;ab&gt;/(\eta_A \eta_B)
why do this conditional expectation must grow?

 

[naima]

I found the algebric origin of the fact that in Bell's theorem we replace 2 by 4/\eta - 2
suppose that a detector has a probability \eta to detect a particle
we have four possibilities when a pair is created:
1) left and right not detected
2) left detected but not right
3) right detected but not left
4) left and right detected
the probabilities are:
(1 - \eta)^2
\eta (1 - \eta)
(1 - \eta)\eta
\eta^2
the observer cannot count the first case events
We have 2) + 3) + 4) = 1 - (1 - \eta)^2) = (2 - \eta) \eta
the conditional probability of 4) knowing that 1) did not occur is
\frac{\eta^2 }{ (2 - \eta) \eta} = \eta/(2 - \eta)
the expected value Bell = \eta/(2 - \eta)&lt;=2 implies Bell &lt;= (4/\eta) - 2