Breaking Bell's Inequality with Monte Carlo Simulations in Python

Bell test: https://en.wikipedia.org/wiki/Bell_test :

> To do away with this assumption it is necessary to detect a sufficiently large fraction of the photons. This is usually characterized in terms of the detection efficiency η [\eta], defined as the probability that a photodetector detects a photon that arrives at it. Anupam Garg and N. David Mermin showed that when using a maximally entangled state and the CHSH inequality an efficiency of η > 2*sqrt(2)/2~= 0.83 is required for a loophole-free violation.[51] Later Philippe H. Eberhard showed that when using a partially entangled state a loophole-free violation is possible for η>2/3~=0.67 which is the optimal bound for the CHSH inequality.[53] Other Bell inequalities allow for even lower bounds. For example, there exists a four-setting inequality which is violated for η>(sqrt(5)-1)/2~=0.62 [54]

CHSH inequality: https://en.wikipedia.org/wiki/CHSH_inequality

/sbin/chsh

Isn't it possible to measure the wake of a photon instead of measuring the photon itself; to measure the wake without affecting the boat that has already passed? And shouldn't a simple beam splitter be enough to demonstrate entanglement if there is an instrument with sufficient sensitivity to infer the phase of a passed photon?

This says that intensity is sufficient to read phase: https://news.ycombinator.com/item?id=40492160 :

> "Bridging coherence optics and classical mechanics: A generic light polarization-entanglement complementary relation" (2023) https://journals.aps.org/prresearch/abstract/10.1103/PhysRev... :

>> This means that hard-to-measure optical properties such as amplitudes, phases and correlations—perhaps even these of quantum wave systems—can be deduced from something a lot easier to measure: light intensity

And all it takes to win the game is to transmit classical bits with digital error correction using hidden variables?

This state is local and "inside" our universe, but we can't observe it. (A good analog for thing that are unobservable from inside the universe are seed of a pseudo-random generator).

The beauty of it, is just realising that Nature's simulator can be purely local and yet not be subjected to Bell Inequalities, but still reproduce the spurious quantum correlations, if you calculate the probabilities.

Violating Bell Inequalities is totally normal when you construct your theory such that Bell Inequalities don't apply.

What if there are no hidden properties per particle, but the combination of specific property values of particles allow for breaking Bell's Inequality?

I.e. what we call 'entanglement', it might not be 'action-at-a-distance', but the simple effect of the interaction of the properties of the two particles as they are generated.

For example, if we have two billiard balls, which are really close together, and we hit them with a third ball simultaneously, their spin will be correlated when we measure it for both balls (without taking into account other factors, i.e. friction, tilting of the table etc). Wouldn't that break Bell's inequality as well? the spins of the two balls will be correlated.

> Everything up to the [spoiler ahead] in this comment is (as far as I can tell) exactly how things work in standard formulations of Bell's inequality. There's nothing weird or crackpot there.

Moreover, to clarify, it's not necessary that the hidden variables can be measurable or that you can set them. So if a system like the one you described must follow the Bell's Inequality if all the other hypothesis are true.

I read the code and it looks like an accurate implementation of the model proposed in the paper.

From the paper you liked:

> However, the expectation values E(X1X2), displayed in (13) contain a factor 1/2, meaning that they do not violate CHSH inequality.

I agree with that part. The model should not violate the Bell's inequality or the equivalent version.

> The agreement with quantum predictions is obtained only after the “photon identification procedure”, which selects, from the raw data, final data samples.

The selection rule is the weird part. It's described in equations 7 and 8.

  x := sign(1 + cos[2(a − φ)] − 2 · r1)

a is the angle of the polarizer

φ is the secret variable that is the angle of the photon. (QM says that this type of entangled photons have no a secret angle, this model assumes that each photon has a hidden variable that is the secret value φ.)

So far so good, this calculation gives the expected result if you assume that φ is chosen from a uniform distribution between 0° and 360°.

  v := r2 |sin[2(a − φ)]|^d (Vmax − Vmin) − Vmax
  selected := (v ≤ V)

With the numbers in your program

  v := r2 |sin[2(a − φ)]|^2 (10 − 0) − 10
  selected := (v ≤ -9.99)

  selected := r2 |sin[2(a − φ)]|^2 ≤ -0.001

Most of the times r2 is not tiny, so most of the times this means that the sine is tiny that means that the secret angle of the photon is almost aligned or almost orthogonal to the polarizer.

So this is a device that can measure the secret angle of the photon. This is not a real device, so it can't be proposed as an alternative explanation of the violation of the Bell's inequality.

You may be wondering why I claim it's not a real device.

If you have a detector of polarization, once you fix the angle 'a', you can't distinguish:

1) Unpolarized light, that is in particular the type of light used in a Bell's inequality test where the state is (|00> + |11>)/sqrt(2) or in other versions (|01> + |10>)/sqrt(2), where 0 is horizontal and v is vertical, or a uniform random values of φ in the model of the paper

2) Light polarized in 45° to the detector's angle, that is like a constant φ in both models.

In both cases, you detect 50% of the photons.

If you use the selection device of this paper,

1) with unpolarized light you will get selections when r2 is very small or when φ is almost paraller or orthogonal to the angle a.

2) with polarized light at 45° you will get selections when r2 is very small

So with polarized light at 45° the number of events will be much smaller than with not polarized light.

In particular if you have the source of not polarized light and the detector, adding a polarizer at 45° in the middle will reduce the number of events in the firs case to 1/4 and in the other to almost 0.

What if the particles have properties that mutate their state after they are sent to Alice and Bob?

Suppose, in the billiards example, that I put a small device into the balls that changes the spin of the ball to some predefined value.

Wouldn't that break the Bell inequalities without action at a distance?

The reason for the breaking would be that the state of the balls would be modified after they are sent to Alice and Bob. It would look like action at a distance without being 'action at a distance'.