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

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

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_1xσ_2yσ_3y|ψ>, σ_1yσ_2xσ_3y|ψ>, σ_1yσ_2yσ_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_1xσ_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:

http://media.physics.harvard.edu/video/?id=SidneyColeman_QMIYF

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 m_x or m_y. Then they compare the measurements and notice these experimental correlations:

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.