# Can synchronization emulate non-locality ?

1. Dec 5, 2015

### Mentz114

I remember reading some time ago of a hypothesis ( I can't find it now) that quantum superpositions could be modelled by oscillating states. So some two valued attribute could be (say) 0 for half the time and 1 for other half ( the value flip-flops).

Individual measurements at random (ish) intervals will yield a random binary sequence. Suppose Alice and Bob receive objects to test from a common source placed btween them, where the two objects share the same flip-flop function on the attribute in question. The probabilities of the 4 possible outcomes will depend on the time and position of Alice and Bobs measurements. One could therefore find, in principle a setup ( times and positions of the measurements) to emulate anything from zero correlation to maximum positive or negative correlation.

I have not attempted to calculate whether any of the probabilist bounds CH, CHSH etc can be broken, but I think the odds are against, given that this clock looks like a local hidden variable.

I can see plenty of difficulties with this, but not the anything that eliminates it certainly. What did I miss ?

2. Dec 5, 2015

### Heinera

Last edited: Dec 5, 2015
3. Dec 5, 2015

### entropy1

How would this scheme conform to Malus' probability correlation?

Last edited: Dec 5, 2015
4. Dec 5, 2015

### Mentz114

I don't understand what 'The measurements that Alice and Bob perform should be random and completely uncorrelated with whatever is produced at the source' means operationally.

I can't decide if it is the same as the coincidence loophole. I need to think about it. Very relevant, thanks.

I see that Wiki article mentions R D Gill (2004) who is a member here.

5. Dec 5, 2015

### Mentz114

I don't know Malus' probabiity correlation.

6. Dec 5, 2015

### entropy1

Sorry, I am not making myself clear. See Malus' law.

7. Dec 5, 2015

### Mentz114

The issue is whether synchronization can possibly account for correlations which at present are attributed by some to non-locality.
Polarizers are not relevant.

8. Dec 5, 2015

### Mentz114

I've done some more calculation and I get an interference term in agreement with the polariser equation.

In the basis of the synchronised attribute $|0\rangle, |1\rangle$ the left and right state vectors are

$|\psi_a\rangle = \phi_a |0\rangle + \phi_a^{*} |1\rangle$ and the same for $|\psi_b\rangle$. The correlation density (terminology?) is
\begin{align}
\rho_{ab} &= |\psi_a\rangle\langle\psi_b| = \left[ \begin{array}{cc} \phi_a \phi_b & \phi_a^{*} \phi_b \\
\phi_b^{*} \phi_a & \phi_a^{*} \phi_b^{*} \\
\end{array} \right]
\end{align}
To get the full space-time dependency let $\phi_a= e^{i(\omega t_a + k\ x_a)}$ and for $\phi_b= e^{i(\omega t_b - k\ x_b)}$. The two eigenvectors of $\rho_{ab}$ are

$v_1=(1, -{e}^{2\,i\,t_b\,w-2\,i\,k\,x_b})/\sqrt{2},\ \ v_2=(1, {e}^{-2\,i\,k\,x_a-2\,i\,t_a\,w})/\sqrt{2}$

The first eigenvalue is zero, so I calculated $\langle v_2|\rho_{ab}|v_2\rangle$ which after some algebra is

$\tfrac{1}{2}\cos\left( 2\,k\,(x_b-x_a)-2\,\omega\ (t_b+t_a)\right)+\frac{1}{2}$

the same value is found for $\langle \psi_a|\rho_{ab}|\psi_a\rangle$. which looks familiar.

I think this shows that there is time and space dependent interference here. The eigenvalue of $v_2$ is $\phi_a \phi_b +\phi_a^{*} \phi_b^{*}$ which looks like a standing wave.

Last edited: Dec 5, 2015
9. Dec 6, 2015

### stevendaryl

Staff Emeritus
I think that it helps to start with a simplified case. Use polarizing filters in the x-y plane that can be rotated to three orientations:
1. The V direction is aligned with the y-axis (and the H direction is aligned with the x-axis).
2. The V direction is at a 120o angle to the y-axis (measured clockwise)
3. The V direction is at a 240o angle to the y-axis (measured clockwise)
In the experiment, Alice and Bob each pick one of the three orientations to measure the polarization of one photon of an EPR twin pair. The results, as predicted by QM, are:
• They each have a 50/50 chance of getting H or V, regardless of their choice of orientations
• If Alice and Bob choose the same orientation, then they get the same result, either $V$ or $H$
• If they choose different orientations, then they get the same result 25% of the time, and different results 75% of the time.
To simulate these results, you would need a pair of computer programs, "R_bob" and "R_alice", with the following properties:
1. You start them off together at some time $t_1$.
2. At time $t_2 > t_1$, Alice inputs her orientation to the program "R_alice": the number 1, 2, or 3.
3. At time $t_3 > t_1$, Bob inputs his orientation to the program "R_bob".
4. At time $t_4 > t_2$, "R_alice" outputs "H" or "V".
5. At time $t_5 > t_3$, "R_bob" outputs "H" or "V".
And you have to make sure that "R_bob" doesn't communicate with "R_alice".

(What isn't covered by such a simulation is the possibility of errors--Alice or Bob might miss some photon)

With a setup like this, you can certainly allow the programs to have built-in clocks, and their outputs could be allowed to depend on the timing. But I don't see how allowing time dependence would help anything.

10. Dec 6, 2015

### Mentz114

Thanks. All this can be compactly examined by looking at interference terms for space and/or time dependent terms. See Ballentine for instance.

The rotating model above predicts the cos2(a-b) rule but where a and b are space-time intervals not angles of rotation of filters.

11. Dec 6, 2015

### stevendaryl

Staff Emeritus
The relevant details of EPR have been extracted in my description, so it's not necessary to consider arbitrary angles. The only relevant numbers are:

1. $cos^2(120) = cos^2(240) = 1/4$
2. $cos^2(0) = 1$
The differences are always 0, 120 or 240.

So confining ourselves to these three possibilities, a simulation of EPR would have to reproduce the results that:
• If Bob and Alice choose the same orientation, then half the time they both get H, and half the time they both get V.
• If Bob and Alice choose different orientations, then 1/8 the time, they both get H, 1/8 the time they both get V, 3/8 of the time, Alice gets H and Bob gets V, and 3/8 of the time, Alice gets V and Bob gets H.
A rotating model doesn't seem like it could reproduce these predictions.

12. Dec 6, 2015

### Mentz114

Yes, it looks very unlikely. But I'll investigate in any case.

Maybe I can make the synchronising field $\phi$ tachyonic ...