Breaking Bell's Inequality with Monte Carlo Simulations in Python

> -0.001

Should be -0.01 You have added an extra 0, but it's not the point. You can pick any V, but the bigger it is the more quantum like the correlation are.

>secret angle

Also called "phase" this is the thing there is to "see" : It has a definite value for a single experiment, but every time you do the experiment it has a different value. It behaves like a random variable and that's what allows you to replicate the behavior of what QM does by generating random numbers. That's the subtlety that makes it so that Bell's theorem don't apply.

>So this is a device that can measure the secret angle of the photon.

It uses the secret angle of the photon, to give you something observable, but doesn't leak info about the state. ("it mixes trajectory space" so that each trajectory behave the same, but trajectory are independent, each trajectory just cycle through all the possible hidden states (like the seeds of a linear congruential generator) ).

For a definite (monte-carlo) trajectory, the photon will be definitely absorbed, or not absorbed, (or maybe absorbed later), but the simulator has a state and knows unambiguously how to evolve it, you as an observer though will have to define measurements more ambiguously (due to Heisenberg uncertainty principle (but that's not the point here) )

One other way to see what we are trying to do is factorizing the QM integral.

In QM you have proba = integral( wavefunction ),

You introduce a random variable and condition on it by writing it as proba = integral( integral( wavefunction | hidden_state) dhidden_state )

The point being that you can be smart in the choice of the hidden_state such that the inner integral behaves classically : You push the quantum correlation to the outside integral.

If you want to calculate the probability, you use monte-carlo for the outside integral. And classical simulation for the inner one.

But once written in such a way, you realise that if you want to simulate a universe (like Nature does it), you don't have to simulate all trajectories : Any one will do, as they are all independent from each other.

From inside the universe because you don't have the initial phase, if you want to calculate the proba you have to do a monte-carlo.

>> secret angle

> Also called "phase" this is the thing there is to "see" : It has a definite value for a single experiment, but every time you do the experiment it has a different value. It behaves like a random variable and that's what allows you to replicate the behavior of what QM does by generating random numbers. That's the subtlety that makes it so that Bell's theorem don't apply.

That's standard local hidden variable theory. Bell's theorem apply.

The problem is that the device that is used in the article gives the wrong prediction for a beam with 50% vertically (φ=0) polarized light and 50% horizontally polarizad light (φ=90°). What is the ratio of selected photons as a function of the angle a?

In the script I already gave you it shows an even stronger argument than CHSH inequality : Convergence (in law) towards the QM probas : It can replicate all the proba given by QM for any alpha,beta polarizer settings, up to epsilon, with epsilon that can be made vanishingly small.

QM breaks CHSH inequality, this replicates the proba of QM therefore it also breaks CHSH.

Of course I'm not banging against a math theorem wall, I just made some leeway to go around, based on the fact that conditional probabilities are not probabilities. Setting the problem such that measurements/observation are defined as a conditional probability (against an unobservable variable) suffice for making Bell theorem not applicable. It offers a whole class of solution to the seemingly paradoxical Bell Inequalities.

That the subtlety of this post-selection scheme, the state is completely local : By construction (L37) sela only depends on particle a, and (L38) selb only depends on particle b.

The measurement of a only depend on sela (and not selb), and the measurement of b only depend on selb (and not sela). There is no exchange of information.

The universe already has given you the observations it needed to give you by line 38. The simulator only used local information to simulate the universe up to this point.

Like in qm once you have written down your measurements, you compare them to count coincidences. Sela just mean you registered a click on detector a, Selb just mean you registered a click on detector b. The logical_and is just you counting the observations as a coincidence or not, aka whether you got a click on both detector simultaneously. You are free to be as non-local as you want here, it is of no importance with regard to the state of the universe, the clicks already happened or not happened.

Ok I think I understand your intention with the code now. Sorry I was wrong before. I think what you're talking about here is what gets called the "detection loophole" in most of the literatire. The idea that if we detect only a small enough fraction of the events then they can be a sufficiently unrepresentative sample that we think we violated a Bell inequality even though the full statistics don't.

This has (in my opinion) been comprehensively addressed already. You can check out the references in the section of the wiki article here

https://en.wikipedia.org/wiki/Bell_test#Detection_loophole

but basically if you detect enough of the possible events in the experiment there is no way for nature to "trick" you in this way, "enough" is 83% for the standard CHSH inequality or 66% if you use a slightly modified. Recent experiments (in the last decade or so) are substantially over the threshold for the detection loophole to be a problem. This is one of the earliest papers where this loophole was closed with space-like seperated detectors from 2015. In this paper they used entangled NV centers in diamond as their qubits of choice, and so essentialy had zero events lost.

https://arxiv.org/abs/1508.05949

This is a second one from the same time. This one uses a more standard setup with photons and worked with about 75% detector efficiency for each party (well above the 66% required)

https://arxiv.org/abs/1511.03189

And this is a third with an efficiency of 78% for Alice and 76% for Bob

https://arxiv.org/abs/1511.03190

I therefore have a new challenge - break the CHSH inequality, while rejecting fewer than 17% of the events, or break the (easier) modified Bell inequality used in papers 2 & 3 while rejecting fewer than a third.

Edit: This is another, more recent paper where they use superconducting qubits and again lose no events

https://www.nature.com/articles/s41586-023-05885-0