The CHSH urn model

[gill1109]

Of course Bell's inequality or the CHSH is a tautology for a 4xN spreadsheet of outcomes. The inequalities are irrelevant for physics http://dx.doi.org/10.1166/qm.2014.1153

This is a straw man argument. In the CHSH urn experiment, there is no 4xN spreadsheet of outcomes. There are four 2xn spreadsheets. That's the whole point.

Heinera, you are not quite right. I would say it like this.

The data from the CHSH urn experiment can be placed in an (N rows) x (4 columns) spreadsheet. Each *row* contains Alice’s setting (“1” or “2”), Bob’s setting (“1” or ‘2”), Alice’s outcome (+/-1), Bob’s outcome (+/-1).
This can be split into four data-sets of sizes N11, N12, N21 and N22, each consisting of pairs of outcomes.
N11 + N12 + N21 + N22 = N

I assume that by "4xN spreadsheet" @minkwe did not mean the 4xN spreadsheet you construct here, but a spreadsheet with four columns, two of which are actual outcomes, and the other two counterfactual outcomes. No need to construct such a spreadsheet when performing the CHSH urn experiment. Only actual outcomes need to be recorded, just as in any other physical experiment.

Hm, maybe. I think that @minkwe is confused. I think he’s thinking of the Nx4 matrix of factual and counterfactual outcomes (no settings) which appears in my Statistical Science paper.

In my simulation there is, in effect,
- an N x 4 spreadsheet of sets of four counterfactual outcomes X1, X2, Y1, Y2
- an N x 2 spreadsheet of settings A, B
- an N x 4 spreadsheet of observed Dara (settings and outcomes) A, B, X, Y

The relationships between them are that X = X_A, Y = Y_B; and the settings are independent of the quadruplets of “potential outcomes”

Obviously, CHSH holds for the counterfactuals
The point is to show that it holds, in expectation value, for the observed data (conditional on the subsample sizes Nij)
That’s all very easy.
The tricky thing is to give probability bounds on “statistical fluctuations”. That’s what I did, in three different ways over the years, in three different papers; and my bounds have since been improved (Peter Bierhorst; and the Delft people; and others).

*****
Added later, response to *next* post:

I don't think it is important how people convinces themselves that a CHSH urn experiment has a long run upper bound of 2 for CHSH expression, and can thus not reproduce the expected results of a quantum experiment. One could be convinced by some mathematical argument, or alternatively by doing a lot of simulations with different distributions of the slips.

The important point is that if you are convinced the statistical upper bound is two for the CHSH urn experiment, you also believe Bell's theorem to be true, because this is Bell's theorem. Any LHV model can be reduced to a CHSH urn experiment, as explained here:

viewtopic.php?f=6&t=489

Agreed!

[Heinera]

I don't think it is important how people convinces themselves that a CHSH urn experiment has a long run upper bound of 2 for CHSH expression, and can thus not reproduce the expected results of a quantum experiment. One could be convinced by some mathematical argument, or alternatively by doing a lot of simulations with different distributions of the slips.

The important point is that if you are convinced the statistical upper bound is two for the CHSH urn experiment, you also believe Bell's theorem to be true, because this is Bell's theorem. Any LHV model can be reduced to a CHSH urn experiment, as explained here:

viewtopic.php?f=6&t=489

[gill1109]

This RPubs document computes the optimal urn in the CHSH urn model.
https://rpubs.com/gill1109/optimalUrn

Just 8 slips of paper: "0000" "0100" "0110" "1110" "0001" "1001" "1011" "1111"; these are the outcomes of the semi-hidden variables x1, x2, y1, y2; outcomes coded 0/1 instead of -1/+1
The code *computes* those 8 four-bit strings.

The code is a bit clumsy because the easy way to create this is using the order x1, y1, x2, y2 (thought of as lying on the corners of a square starting at top left hand corner, and going clockwise). The urn is optimal for getting high correlations in the 21, 11, 12 settings; high and negative in the 22 setting. For large N those four correlations will be close to +/- 0.5.

Recall: in the model one picks one slip of paper. One independently picks a and b, = 1 or 2. One then reads off the chosen xa and yb.
You can think of half of the slip of paper going to Alice, the other half to Bob; and Alice and Bob can choose their settings independently of one another.
But they don't have to.

Put the slip back in the urn, repeat.