[ExI] Many Worlds, realism and Leggett's Inequality

[Stuart LaForge]

On 2026-03-23 07:23, John Clark via extropy-chat wrote:
> I asked Gemini to explain what the experimentally derived fact that
> Leggett's inequality is violated tells us that Bell's Inequality does
> not , and what implications that has on the Many Worlds interpretation
> of quantum mechanics. This is what he she or it said:
> 
> ===
> 
> When experiments showed that Bell's inequality is violated, it proved
> that at least one of these must be false. Most physicists opted to
> ditch locality, leading to the acceptance of "quantum non-locality"
> (entanglement). However, this left the door open for "non-local
> realism"—the idea that particles _do_ have definite properties, but
> they are coordinated by some instantaneous, hidden signal.
> 
> -------------------------

Well that is what the wavefunction is, isn't it? A hidden signal that 
coordinates particle properties in a definable pattern? I would imagine 
that quantum states from multiple systems that become entangled with one 
another during decoherence would share some global or universal hidden 
variable; a distinct master phase shift, for example, that could 
distinguish one Everett branch from another in a well-ordered manner.

> 
> LEGGETT’S INEQUALITY: THE ATTACK ON REALISM
> 
> The violation of Leggett’s inequality (first observed experimentally
> in 2007) tells us something much more radical:
> It isn't just about speed: Even if we allow for "spooky action at a
> distance" (non-locality), we still cannot explain the experimental
> results if we assume that particles have definite, pre-existing
> states.


Yes, the effect is even more pronounced and "unrealistic" when 3 or more 
particles are entangled together. Take the Greenberger–Horne–Zeilinger 
(GHZ) triplet state. If we prepare a set of three particles using a 
Gerlach-Sterns device to be in a GHZ-state with all their spins aligned 
in the same z-orientation, then the three particles displays the 
following spin correlations when all are measured in the x orientation 
or when one is measured in the x-orientation and the other two are 
measured in the y-orientation:

{Ax}{Bx}{Cx} = -1
{Ax}{By}{Cy} = +1
{Ay}{Bx}{Cy} = +1
{Ay}{By}{Cx} = +1

Here Ax is the spin of particle A in the x-direction and can be +1 for 
spin up and -1 for spin down. Similarly for the other variables. In 
experiments, the products of their spins always follows the given 
pattern as predicted by QM. The four equations above are true.

Now let's assume reality in the sense that the spins of particles A, B, 
and C in the x and y-directions are "real" and fixed in a definite state 
of spin up or spin down before they are measured.

If you pay close attention to those equations, then you will notice that 
there is no way to pre-assign spin states to A, B, C in the X and Y 
directions to make all three equations true at the same time.

For example, let's say that you preassign A, B, and C to spin down in 
the x-direction leading to the first of the only two possible scenarios.
Substituting -1 in for Ax, Bx, and Cx yields:

{-1}{-1}{-1} = -1 for the first equation and for the rest we get
{-1}{By}{Cy} = +1
{Ay}{-1}{Cy} = +1
{Ay}{By}{-1} = +1

Dividing all the equations by -1 on both sides yields:
{-1}{-1} = +1
{By}{Cy} = -1
{Ay}{Cy} = -1
{Ay}{By} = -1



Here you can clearly see that the spins of all three particles in the 
y-direction must be simultaneously be different (opposites) from one 
another, but that is not possible when you only have the two states of 
up or down to choose from. You can't have three numbers that are all 
inverses of one another without two of them being the same number. This 
contradicts out starting premise all three spins were down in 
x-direction.

So what about the second possible scenario with two of A, B, and C being 
spin up in the x-direction and only one of them being spin down in x.

{-1}{+1}{+1} = -1
{-1}{By}{Cy} = +1
{Ay}{+1}{Cy} = +1
{Ay}{By}{+1} = +1

Dividing out the factor {+1} as redundant gives

{-1}{+1} = -1
{-1}{By}{Cy} = +1
{Ay}{Cy} = +1
{Ay}{By} = +1

So now we divide both sides of equation 2 by {-1} to yield
{-1}{+1} = -1
{By}{Cy} = -1
{Ay}{Cy} = +1
{Ay}{By} = +1

And we can clearly see that the spins {By} and {Cy} must  be different 
from one another, but simultaneously the same as the spin {Ay}. Another 
contradiction.

One can see that by symmetry arguments, it would not matter which 
particle A, B, or C was the one that started out spin down in the 
x-direction, each and every time, a contradiction is reached.

Therefore the assumption of the spins of particles having a real value 
prior to measurement is proven false by contradiction.

Q.E.D.

Stuart LaForge