## Followup to minkwe's mutual information analysis

Foundations of physics and/or philosophy of physics, and in particular, posts on unresolved or controversial issues

### Followup to minkwe's mutual information analysis

In this post (original thread is locked):

viewtopic.php?p=10257#p10257

minkwe derives no mututal information between the outcomes. However the analysis appears incorrect. Consider the case that the settings are 0 and pi/8. Then we have from QM (minkwe gives 1/4 for all of these):

P-- = 0.5 * cos^2(pi/8) = 0.42678
P++ = 0.5 * cos^2(pi/8) = 0.42678
P-+ = 0.5 * sin^2(pi/8) = 0.07322
P+- = 0.5 * sin^2(pi/8) = 0.07322

These produce a non-zero mutual information between the outcomes. All the other angle combinations also yield non-zero mutual information.

I wrote a simulation to prove this. I implemented the quantum joint solution with the standard angle sets, and produced settings (chosen randomly) and outcome sequences (by sampling the distributions with 10000000 samples). I verified that the CHSH is violated and calculated the mutual informations. Here is the result (S means setting, O means outcome, A and B are the two sides):

-----
Generate sequences using the quantum joint prediction.

Calculate the expectation values and the CHSH metric.
S greater than 2 is a violation of the CHSH inequality.
e1 = 0.707306, e2 = -0.707511, e3 = 0.706807, e4 = 0.707812
S = 2.829437

MI SB-OA 0.000000
MI SA-OB 0.000000
MI SA-SB 0.000000
MI OA-OB 0.092227
MI SA-OA 0.000000
MI SB-OB 0.000000
-----

It's intuitively obvious to me that the two outcome streams will have a non-zero mutual information. After all, the streams are correlated, otherwise CHSH would not be violated.

Perhaps minkwe goes wrong by averaging together the results for different angle combinations.

The interesting thing for me is that there is no mutual information involving settings despite published claims that setting and outcome dependence are needed for violating CHSH. What's going on? Thinking simply, the outcomes at B depend on the angle difference, and hence depend on each setting, however, we do not see mutual information between SA and OB. I have an idea about this and will report further if it pans out.
local

Posts: 86
Joined: Mon Aug 05, 2019 1:19 pm

### Re: Followup to minkwe's mutual information analysis

This was a good post.

local wrote:The interesting thing for me is that there is no mutual information involving settings despite published claims that setting and outcome dependence are needed for violating CHSH. What's going on?

I'm not sure what you mean here. The claims are that for a local hidden variable model dependence between settings and the hidden variable is needed for violating CHSH. This is not something you can analyze from data alone, since the hidden variable is, well, hidden.
Heinera

Posts: 718
Joined: Thu Feb 06, 2014 1:50 am

### Re: Followup to minkwe's mutual information analysis

Thank you for your reply and, apparently, agreeing with my analysis. I'll say again, even in QM: Thinking simply, the outcomes at B depend on the angle difference, and hence depend on each setting, however, we do not see mutual information between SA and OB.

Do you have an explanation for that?

(As an aside, I see QM as subject to all the no-go theorems that LHV must face. But that's another debate.)
local

Posts: 86
Joined: Mon Aug 05, 2019 1:19 pm

### Re: Followup to minkwe's mutual information analysis

local wrote: I'll say again, even in QM: Thinking simply, the outcomes at B depend on the angle difference, and hence depend on each setting, however, we do not see mutual information between SA and OB.

No, it doesn't actually. In QM, the outcomes at B will be randomly 50/50 no matter what the angle difference is.
Heinera

Posts: 718
Joined: Thu Feb 06, 2014 1:50 am

### Re: Followup to minkwe's mutual information analysis

Thank you for your reply. I'll come back to you on that later. Gotta go out for a few hours.
local

Posts: 86
Joined: Mon Aug 05, 2019 1:19 pm

### Re: Followup to minkwe's mutual information analysis

local wrote:In this post (original thread is locked):

viewtopic.php?p=10257#p10257

minkwe derives no mututal information between the outcomes. However the analysis appears incorrect. Consider the case that the settings are 0 and pi/8. Then we have from QM (minkwe gives 1/4 for all of these):

P-- = 0.5 * cos^2(pi/8) = 0.42678
P++ = 0.5 * cos^2(pi/8) = 0.42678
P-+ = 0.5 * sin^2(pi/8) = 0.07322
P+- = 0.5 * sin^2(pi/8) = 0.07322

I think you misunderstood my analysis. In an experiment, we do not have just 2 settings. There are 4 settings, 2 at each station, with random switching. The analysis I presented is for the case in which the settings are $a \in \{ 0, \frac{\pi}{4} \}$, $b \in \{ \frac{\pi}{8}, \frac{3\pi}{8} \}$. Again note that there are 4 settings with 2 randomly switched on either side. For that case, the probabilities above, are 1/4.

Of course, if you have just two settings with no random switching, then you end up with a list of pairs in which the first value is always the same. Thus, Alice's setting, outcome data looks like:
a x
0 +1
0 -1
0 -1
0 +1
...

and Bob's data looks like
b y
pi/8 +1
pi/8 -1
pi/8 -1
pi/8 +1
...
In this case, the mutual information I(a,b) = 1 (settings dependence) and of course the mutual information between outcomes I(x, y) won't be zero. But this is not the analysis I am doing. I am evaluating the experiment as a whole, in which there is random switching between settings. In any case, there could still be an error in my analysis but I don't think it is what you identify.

I wrote a simulation to prove this. I implemented the quantum joint solution with the standard angle sets, and produced settings (chosen randomly) and outcome sequences (by sampling the distributions with 10000000 samples). I verified that the CHSH is violated and calculated the mutual informations. Here is the result (S means setting, O means outcome, A and B are the two sides):

-----
Generate sequences using the quantum joint prediction.

Calculate the expectation values and the CHSH metric.
S greater than 2 is a violation of the CHSH inequality.
e1 = 0.707306, e2 = -0.707511, e3 = 0.706807, e4 = 0.707812
S = 2.829437

MI SB-OA 0.000000
MI SA-OB 0.000000
MI SA-SB 0.000000
MI OA-OB 0.092227
MI SA-OA 0.000000
MI SB-OB 0.000000
-----

It's intuitively obvious to me that the two outcome streams will have a non-zero mutual information. After all, the streams are correlated, otherwise CHSH would not be violated.

Perhaps minkwe goes wrong by averaging together the results for different angle combinations.

That is what I thought too! I expected to find non-zero mutual information until I did the analysis. So where exactly is the analysis going wrong. The "non-zero" MI OA-OB you obtain above is based on the full dataset with all the different angle combinations so I don't think it is an error to average together the results for different angle combinations. But note that MI OA-OB 0.092227 means that Bob can obtain 0.092227 bits (assuming you used log2) of information about Alice's outcome from his outcomes without ever seeing Alice's outcomes. Could you share the 4 settings you used in your QM simulation?

The interesting thing for me is that there is no mutual information involving settings despite published claims that setting and outcome dependence are needed for violating CHSH. What's going on?

There no settings dependence because you randomly switched the settings. Similarly, if an experiment claims to have randomly switched the settings, there shouldn't be any settings dependence.

Thinking simply, the outcomes at B depend on the angle difference, and hence depend on each setting, however, we do not see mutual information between SA and OB. I have an idea about this and will report further if it pans out.[/quote]
I do not doubt you here, since I did the same using the equations in your paper, last month (at your suggestion).
minkwe

Posts: 1151
Joined: Sat Feb 08, 2014 10:22 am

### Re: Followup to minkwe's mutual information analysis

Are you seeing what these authors saw?
https://arxiv.org/abs/1606.00784
localyokel

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Joined: Thu Oct 24, 2019 5:49 pm

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