## Bell Theorem without inequalities

### The GHZM argument

I had discussed in the past about Bell’s theorem and his inequality for a local realistic theory. Also I have discussed the impossible
noncontextual coloring game of Kochen and Specker. The Kochen-Specker theorem is very striking, but it has a major weakness: it
cannot be put to an experimental test. Why? Because the coloring game can succeed
when there are errors in aligning the measurement directions and no experiment
is perfect. In other words, it is not a robust result. Bell
inequalities are amenable to experimental verification and supporters of local
realism argue then about experimental loopholes (I’ll cover them in future
posts). It would be nice if there is a robust argument like Bell ’s
theorem but just as compelling as the Kochen-Specker theorem.

Interestingly such an argument exists and was introduced by
Greenberger, Horne, Zeilinger, and Mermin and is about three particles in a particular entangled state |ψ> (up to a
normalization coefficient):

|ψ> = |+>

_{1}|+>_{2}|+>_{3}- |->_{1}|->_{2}|->_{3}
where

|+> = 1

0

|-> = 0

1

eigenvectors of the σ

_{z}the operator measuring spin on the z axis:
σ

_{z}= 1 0
0 -1

σ

_{x}= 0 1
1 0

σ

_{y}= 0 -i
i 0

Simple matrix multiplication shows that:

σ

_{x}|+> = |-> σ_{x}|-> = |+>
σ

_{y}|+> = i |-> σ_{y }|-> = -i |+>
σ

_{z}|+> = |+> σ_{z }|-> = - |->
Using those identities compute: σ

_{1x}σ_{2y}σ_{3y}|ψ>, σ_{1y}σ_{2x}σ_{3y}|ψ>, σ_{1y}σ_{2y}σ_{3x}|ψ> and convince yourself that it is equal with |ψ>
Then compute σ

_{1x}σ_{2x}σ_{3x}|ψ> and see that is equal with -|ψ>
From the commutation rule of Pauli matrices it is easy to
see that:

(σ

_{1x}σ_{2y}σ_{3y})( σ_{1y}σ_{2x}σ_{3y})( σ_{1y}σ_{2y}σ_{3x}) = -( σ_{1x}σ_{2x}σ_{3x})
So now for the local realism contradiction:

Measure at each particle spin on x or y axis without
disturbing other particles (σ

_{x }or σ_{y}) and call the measurements m_{x}or m_{y}
From σ

_{1x}σ_{2x}σ_{3x}|ψ> = -|ψ> we get:
m

_{1x}m_{2x}m_{3x }= -1
However from σ

_{1x}σ_{2y}σ_{3y}|ψ> = σ_{1y}σ_{2x}σ_{3y}|ψ> = σ_{1y}σ_{2y}σ_{3x}|ψ> = |ψ> we get:
m

_{1x}m_{2y}m_{3y }= +1
m

_{1y}m_{2x}m_{3y }= +1
m

_{1y}m_{2y}m_{3x }= +1
Multiplying the last three equations and using the fact that
the square of a measurement is always 1 (because m could be only +1 or -1) yields:

m

_{1x}m_{2x}m_{3x }= +1 CONTRADICTION
So what is going on? Late Sidney Coleman has a famous
lecture:

**Quantum Mechanics in your face**where he very humorously explains all this:
With a video

**I will quickly outline Coleman’s explanation:**__spoiler alert__
Perform an experiment in which three people get one of the ½
spin particles in |ψ>. They randomly measure the spin on either x or y axis recording
m

_{x}or m_{y}. Then they compare the measurements and notice these**correlations:**__experimental__
m

_{1x}m_{2x}m_{3x }= -1
m

_{1x}m_{2y}m_{3y }= +1
m

_{1y}m_{2x}m_{3y }= +1
m

_{1y}m_{2y}m_{3x }= +1
Any attempt to explain them using

**local realism**fails because if local realism holds, each measurement is__causally independent of the others__(locality), and each value “m”__has a definite value prior to measurement__(realism). Only then we can multiply the last three equations arriving at a contradiction with the first one.