We all agree thatSzandSxcannot be measured simultaneously. However, for a large number
of spin 1/2 particles, we can assign a fraction of them to havethe following property
- IfSzis measured, we obtain + ̄h/2 with certainty;
- IfSxis measured, we obtain− ̄h/2 with certainty.
What this means is thatif we measureSzon one of these particles, we agree not to measureSxon
that same particle, andif we measureSxon another one of these particles, we agree not to measure
Sxon that particle. In this way, we are never led to actually measureSzandSxsimultaneously,
since of course, we agreed that we cannot. We denote a particle of this type by (z+,x−). Even
though this model is very different from quantum mechanics, itcan actually reproduce the quantum
measurements ofSzandSx, provided there are equal numbers of particles of type (z+,x+) as there
are particles of type (z+,x−).
Next, let us see how this model can reproduce the predictionsof the spin-singlet correlated
measurements. For a particular pair, there should be a perfect match between particles 1 and 2.
For example,
1 : type (z+,x−) ←→ 2 : type (z−,x+) (17.37)The results of correlations in Table 1 can be reproduced if particle 1 and 2 are matched as follows,
particle 1 particle 2
(z+,x−) ←→ (z−,x+)
(z+,x+) ←→ (z−,x−)
(z−,x+) ←→ (z+,x−)
(z−,x−) ←→ (z+,x+) (17.38)with equal populations, 25% for each pair. The key assumption, as always with hidden variable
theory, is that the results measured byAarepredeterminedindependently ofB’s choice of what
to measure.
So far, correlated measurements involving 2 directions canbe accounted for by a theory of
hidden variables. But new things happen when we consider correlated measurements of 3 spins. To
do this, we consider again a problem of spin 1/2, but we now measure in three directions, indicated
by three unit vectorsa,b,c, which, in general, need not be orthogonal to one another.
Extending the above analysis on two measurements, we now assume that the particles belong
to a definite type (a+,b−,c+). This means that on any given particle, one decides to measure
eithera·S, orb·Sorc·S, but never more than one of these for any given particle. The eigenvalues
are then given by the±assignments. Now, to ensure conservation of total angular momentum,
there must again be a perfect match between the spins measured on particles 1 and 2. The particle
pair must then be a member of one of eight types, given this matching. In hidden variable theory,
these eight types are mutually exclusive, and the corresponding sets are mutually disjoint. They