QuantumPhysics.dvi

are listed in the table below,

population particle 1 particle 2

N 1 (a+,b+,c+) (a−,b−,c−)

N 2 (a+,b+,c−) (a−,b−,c+)

N 3 (a+,b−,c+) (a−,b+,c−)

N 4 (a−,b+,c+) (a+,b−,c−)

N 5 (a+,b−,c−) (a−,b+,c+)

N 6 (a−,b+,c−) (a+,b−,c+)

N 7 (a−,b−,c+) (a+,b+,c−)

N 8 (a−,b−,c−) (a+,b+,c+) (17.39)

Needless to say, each population number is positiveNi≥0 fori= 1,···,8. Now, we consider pairs
of measurements by observersAandBin the following fashion.

SupposeAfindsa·S 1 to be +, andBfindsb·S 2 to be + as well. This can happen if the pair
of particles belongs to type 3 or to type 5. Hence the number ofparticles found for this particular
measurement will beN 3 +N 5. We can associate a probability with this particular measurement,

P(a+,b+) = (N 3 +N 5 )/Ntot Ntot=

∑^8

i=1

Ni (17.40)

In a similar manner, we have

P(a+,c+) = (N 2 +N 5 )/Ntot

P(c+,b+) = (N 3 +N 7 )/Ntot (17.41)

Now, sinceN 2 ,N 7 ≥0, we find that there must be an inequality obeyed by these probabilities,

P(a+,b+)≤P(a+,c+) +P(c+,b+) (17.42)

or more generally,

P(aα,bβ)≤P(aα,cγ) +P(cγ,bβ) (17.43)

whereα,β,γnow take any values±. These are the Bell inequalities for this system. They follow
from assigning an independent reality to the spin measured in each of the directionsa,b, andc,
which in turn follows from Einstein’s locality argument.

17.10Quantum predictions for Bell’s inequalities

We shall here calculate those same probabilitiesP(aα,bβ) etc. using the standard rules of quantum
theory. The state|Φ〉, being a spin singlet, may be represented in either one of thefollowing three