## 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 σz the operator measuring spin on the z axis:

σz = 1      0
0    -1

Let also σx and σy be the Pauli matrices corresponding to measuring spin on x and y axis:

σx = 0     1
1     0

σy = 0    -i
i      0

Simple matrix multiplication shows that:

σx|+> =   |->                   σx|-> =    |+>
σy|+> = i |->                   σ|-> = -i |+>
σz|+> =   |+>                  σ|-> = -  |->

Using those identities compute: σ1xσ2yσ3y|ψ>, σ1yσ2xσ3y|ψ>, σ1yσ2yσ3x|ψ> and convince yourself that it is equal with |ψ>

Then compute σ1x σ2x σ3x|ψ> and see that is equal with  -|ψ>

From the commutation rule of Pauli matrices it is easy to see that:
1x σ2y σ3y)( σ1y σ2x σ3y)( σ1y σ2y σ3x) = -( σ1x σ2x σ3x)

So now for the local realism contradiction:

Measure at each particle spin on x or y axis without disturbing other particles (σx or σy) and call the measurements mx or my

From σ1x σ2x σ3x|ψ> = -|ψ> we get:

m1x m2x m3x = -1

However from σ1xσ2y σ3y|ψ> = σ1y σ2x σ3y|ψ> = σ1y σ2y σ3x|ψ> = |ψ> we get:

m1x m2y m3y = +1
m1y m2x m3y = +1
m1y m2y m3x = +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:

m1x m2x m3x = +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 spoiler alert I will quickly outline Coleman’s explanation:

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 mx or my. Then they compare the measurements and notice these experimental correlations:

m1x m2x m3x = -1
m1x m2y m3y = +1
m1y m2x m3y = +1
m1y m2y m3x = +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.