[HN Gopher] Breaking Bell's Inequality with Monte Carlo Simulati...
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Breaking Bell's Inequality with Monte Carlo Simulations in Python
Author : Maro
Score : 99 points
Date : 2024-09-08 15:28 UTC (3 days ago)
(HTM) web link (bytepawn.com)
(TXT) w3m dump (bytepawn.com)
| danwills wrote:
| "a talented college physics student can do it"
|
| I'm afraid I don't qualify for being able to do that, but I feel
| like I'm tantalizingly close to understanding this overall - but
| I'm finding it hard to understand why the lower-right "TT"
| quadrant is transposed in the S=2.828 example (the red box in the
| diagram). Maybe it's obvious if one understands it better?
| n4r9 wrote:
| It requires a chunk of linear algebra to understand, but the
| Wikipedia page has a slightly more detailed explanation:
| https://en.wikipedia.org/wiki/Bell%27s_theorem#Theorem
|
| It's related to the fact that the expected value of A_1 tensor
| B_1 is negative 1/sqrt(2), whilst the expected value of all
| other tensor products are positive 1/sqrt(2).
| jahnu wrote:
| The explanation and table in the Simple English page for this
| helped me grasp it better. (Although the diagram using green
| dots only confuses :) )
|
| Greene's book is a fantastic read too!
|
| https://simple.wikipedia.org/wiki/Bell%27s_theorem
| gus_massa wrote:
| > _Although the diagram using green dots only confuses :)_
|
| It's very confusing. In particular it does not say that the
| box have 3 doors until the middle of the explanations.
| Also, I don't find the example very similar to the Bell's
| Inequality.
|
| Moreover, I expect in a quantum system that when both open
| the same door they get the same result (or the oposite) so
| in a quantum system I expect that when both open the same
| door they get 100% (or 0%) agreement, so insted of 50% I
| expect 1/3 * 100% + 2/3 * 50 % = 66% (or 1/3 * 0% + 2/3 *
| 50 % = 33%).
|
| Anyway, in some versions of the Bell's Inequality the doors
| of the boxes are "misalignment" so on box has white-gray-
| black doors and the other has another ser of colors. let's
| say creme-pink-brown doors. You never have a 100% or 0% of
| coincidences of the results.
| n4r9 wrote:
| This looks like some kind of variant of Bell's Theorem.
| I've not seen it before, but it reminds me of the GHZ
| inequality [0]
|
| [[EDIT - actually I take that back. The GHZ inequality
| refers to three systems whereas your link refers to three
| measurement choices]].
|
| I don't think your link gives a derivation of the quantum
| correlations beyond "Quantum physics says that half the
| time they should get a match".
|
| [0] https://en.wikipedia.org/wiki/Bell%27s_theorem#GHZ%E2%8
| 0%93M...
| Maro wrote:
| Hi, I'm the guy who wrote the article.
|
| In the article, I first show how to "break" the Bell inequality
| without making a reference to any complicated math or Physics,
| this is the section "Breaking the Bell inequality with non-
| local information", which uses the dice roll example. This is
| on purpose, for pedagogical reasons, and this is why the Python
| approach imo is so useful to demonstrate this whole thing: the
| key idea is, to break the inequality, you need to "peek" at the
| other side.
|
| Then, the next mental step is simply the statement that, in
| "real life", you can prepare a composite system (eg. 2 photons
| modeled as 2 qubits) that you can seperate (modeled as the
| split() function in Python), you can send the 2 parts to two
| different observers, they use a certain measurement setup, and
| the whole game is played, statistic computed, etc. and then you
| get this value 2.82 (which breaks the Bell inequality)! So
| somehow, the 2 qubits are doing that we can only model [in
| Python] as peeking!
|
| The actual derivation of how to get that 2.82 is, in some
| sense, almost like a a detail. I think with this approach, even
| a non-physicist can understand what this whole argument is
| (=Bell's genius).
|
| "a talented college physics student can do it" - I'm a
| Physicist, but I'm not working as a Physicist, and I was able
| to derive all the numbers in that table by hand with pen &
| paper directly. I figured if I can do it 15 years out of
| school, so can a talented college physics student!
|
| The next article will be that derivation [of the raw
| probabilities], I just need to transcribe it from my notebook
| to Latex and clean it up. If you want to see the original
| notes:
|
| https://photos.app.goo.gl/sqxLnEhyeZTDD7oA6
| n4r9 wrote:
| This is closely related to my PhD. It was many years ago but if I
| remember rightly there is no need for the assumption of
| determinism - Bell Inequalities hold just as well for random
| local hidden variables.
|
| Simulating the correlations with computer programs is an
| interesting idea, partly because it challenges to those who still
| believe in a "local" reality to demonstrate Bell Inequality
| violations in distributed classical computer systems. Back in the
| day there was a crackpot researcher named Joy Christian who kept
| publishing repetitive papers in the belief that geometric
| algebras provided a counterexample (it looks like he's still
| going strong! [0]). Of course, there's nothing about geometric
| algebras that cannot be modelled in a computer program, so in
| principle Christian should have been able to demonstrate Bell
| violations in a distributed scenario. Needless to say, this
| hasn't happened even though it would be a momentous breakthrough
| in the foundations of physics.
|
| [0] https://ieeexplore.ieee.org/document/9693502
| eigenket wrote:
| You know something has gone horrifically badly when a paper
| begins with
|
| > This reply paper should be read as a continuation of my
| previous reply paper [1], which is a reply published in this
| journal to a previous critique of one of my papers
|
| We're way too deep in replies now, and anyone who values their
| time should get out now.
| 0cf8612b2e1e wrote:
| On the other hand, academics slap fights are magnificently
| petty to behold. In any dispute the intensity
| of feeling is inversely proportional to the value of the
| issues at stake. That is why academic politics are so bitter.
| n4r9 wrote:
| The conclusion reads like someone who can't admit they're
| wrong on Reddit:
|
| > The common defect in the critiques [2], [6], and [14] is
| that, instead of engaging with the original quaternionic
| 3-sphere model presented in my papers [1], [7]- [11] using
| Geometric Algebra, they insist on criticizing entirely
| unrelated flat space models based on matrices and vector
| "algebra." This logical fallacy by itself renders the
| critiques invalid. Nevertheless, in this paper I have
| addressed every claim made in the critique [6] and the
| critiques it relies on, and demonstrated, point by point,
| that none of the claims made in the critiques are correct.
| I have demonstrated that the claims made in the critique
| [6] are neither proven nor justified. In particular, I have
| demonstrated that, contrary to its claims, critique [6] has
| not found any mistakes in my paper [7], or in my other
| related papers, either in the analytical model for the
| singlet correlations or in its event-by-event numerical
| simulations. Moreover, I have brought out a large number of
| mistakes and incorrect statements from the critique [6] and
| the critiques it relies on. Some of these mistakes are
| surprisingly elementary.
| c3534l wrote:
| I thought that was satire. I can't believe someone
| actually wrote that.
| marcosdumay wrote:
| That looks like a person attacked by a troll in a
| position of power.
|
| But then, I don't want to read the actual claims.
| Maro wrote:
| Hi, author of the article here.
|
| Regarding determinism, I think the reason the assertion is "no
| deterministic local hidden.." is that, you need to break both
| the deterministic and locality assumption. However there is a
| nuance, which is, do you need to break both properties to..
|
| (a) break the Bell inequalities, or, to
|
| (b) reproduce quantum mechanics..
|
| which is not exactly the same thing.
|
| For example, in my toy simulation framework, this [1] simple
| setup --- where Alice's two measurement devices always return
| +1, and Bob's two measurement devices are conditioned on
| Alice's returned value, without any randomness --- breaks the
| Bell-inequalities at S=4, but:
|
| (1) it's not physical, because it also breaks the Tsirelson
| bound (4 > 2.82), ie. you can't actually achieve this with any
| known real-world physical system
|
| (2) it's deterministic in the sense that the code does not call
| `random()`
|
| (3) but from the perspective of Bob, who "calls" the
| measurement function, it would still appear random, since it
| depends on whether Alice measures H or T, which was the outcome
| of a random coin flip; so whether we consider this random is
| quite nuanced..
|
| So the above is an interesting thought/Python experiment for
| what it takes to break the Bell inequalities. Then, if we
| modify the code to reproduce quantum mechanics (for which the 2
| qubits stand in), which is the code shown in the original post,
| in that case we cannot even avoid calling `random()`, because
| the "first" to measure their qubit must also get +1 and -1 with
| equal probabilility, so the theory cannot be deterministic.
|
| [1]
| https://gist.github.com/mtrencseni/de13f766911aaaf5bfd5d4636...
| n4r9 wrote:
| Yes I could have worded that better!
|
| So... what you have here is a deterministic non-local hidden
| variable model which violates Bell Inequalities. The reduced
| probabilities at Bob's end might look random to him, but
| fundamentally the measurement outcomes are determined by
| Alice and Bob's measurement choices. All good.
|
| You also know that any deterministic local hidden variable
| model must obey Bell Inequalities.
|
| What I'm saying is that _any_ local hidden variable model
| must obey Bell Inequalities. You cannot increase the value of
| S by relaxing determinism.
|
| So actually it's kind of a distraction to bring in
| determinism. Either you have local hidden variables - which
| obey Bell Inequalities - or you allow non-local hidden
| variables - in which case Bell Inequalities can be violated.
| Locality is the key assumption.
| GistNoesis wrote:
| The Bell's Inequalities are a test for the capacity for inductive
| reasoning of the pupil. If the pupil succeed he is not to be
| admitted to join the ranks of quantum physicist.
|
| You usually show a pupil the problem with classical
| probabilities, and show that you can't violate Bell's
| Inequalities, then you show that Quantum Mechanics managed to
| replicated the observed probabilities using a non-local way, and
| therefore you conclude that the world is non-local.
|
| But this logic doesn't stand. You need to use inductive reasoning
| to see it through. Ask yourself the question, what change would
| it take to your theory to make it local and still replicate the
| observed probabilities (and still look reasonable).
|
| Solve the riddle (it's quite beautiful once you see it :) ) and
| you will be rewarded with the awesome title of crackpot
| physicist, pitted against other dubious crackpot physicist each
| convinced their loopholes are the ones and only.
| gus_massa wrote:
| > _and therefore you conclude that the world is non-local._
|
| No, Bell's inequality has a few sensible assumtions, like
| locality. The conclusion is that at least one of them is wrong
| and real world is a sensible one :(. By the way, there is this
| crazy thing call QM that nobody likes but gives accurate
| results.
| GistNoesis wrote:
| Just because there is a way, doesn't make it the only way.
|
| >but gives accurate results.
|
| Giving accurate results is missing the point.
|
| Hint: The point is understanding how nature's does it.
|
| Here is the Chesterton's fence implied by Bell's Inequality :
|
| Lemma: There exist a local classical simulator that allows to
| simulate a universe that behaves according to the
| probabilities of QM.
|
| Corollary : we can simulate "fast" a universe which behaves
| (in law) exactly like our universe.
|
| Nota Bene : This doesn't mean we can compute QM probabilities
| fast, (we can't), although one way of computing them would be
| to use Montecarlo estimation on various instances of universe
| simulations.
|
| The question is not whether to lift the fence, or how to lift
| the fence, the question is how are numerical biological
| instabilities handled.
| Joker_vD wrote:
| > Giving accurate results is missing the point.
|
| No, that's the central point of modern (starting with
| Newton) physical science. In fact, I'd argue that's the
| _main_ reason for the astonishing advances of the physics
| in mere 300 years: people stopped bothering too much about
| philosophical underpinning of reality and started to
| fucking measure the reality instead, as precisely and
| accurately as they could and then some. Fresnel 's optics
| won over Newton's not because of its superior philosophical
| merits (it needs luminiferous aether to be a perfectly
| rigid incompressible solid, after all), but simply because
| it very accurately described light's interference,
| diffraction, all kinds of refraction and the accompanying
| polarization, and also dispersion, all in one nice, self-
| contained package. That's what mattered, not the
| ridiculousness or reasonableness of proposition that light
| corpuscles have poles and can experience fits of easy
| transmission/reflection.
|
| > Hint: The point is understanding how nature's does it.
|
| By being itself, how else? /s
| eigenket wrote:
| > You usually show a pupil the problem with classical
| probabilities, and show that you can't violate Bell's
| Inequalities, then you show that Quantum Mechanics managed to
| replicated the observed probabilities using a non-local way,
| and therefore you conclude that the world is non-local.
|
| If you do this you're doing a bad job at being a teacher.
|
| The way the argument should go is you start with a list of
| assumptions (of which locality is one), derive Bell's
| inequality from them, and determine that as Bell's inequality
| seems to be false in real experiments at least one of your
| assumptions was wrong. Then you can talk about quantum
| mechanics and explain which of these assumption are broken in
| quantum mechanics. If you have time you can have fun talking
| about different interpretations of quantum mechanics because
| (e.g.) Everettian Many Worlds is completely local, but still
| produces predictions matching quantum mechanics (and therefore
| breaks Bell's inequality).
| GistNoesis wrote:
| >If you do this you're doing a bad job at being a teacher.
|
| If this wasn't sufficiently clear, I am not a teacher, I am a
| crackpot physicist.
|
| Hint : Listing the assumptions doesn't work. This is what I
| call the three-card monte argument. The ball is not under one
| of the three goblets, the ball is in the sleeve of the
| magician.
| eigenket wrote:
| > Hint : Listing the assumptions doesn't work
|
| Why not?
|
| I could see that it might not if you are not clear about
| your assumptions
| GistNoesis wrote:
| It's circular reasoning, hidden in the definition of your
| assumptions. By defining not clearly what a measurement
| is and observations are.
|
| You must let the cat step out of the box your definitions
| put you in.
|
| You have infinite freedom in your choices of definitions,
| listing assumptions is creating a false dichotomy.
| Especially when doing so conclude to exclude the most
| probable assumption : Locality.
|
| Preserve locality, and find another self consistent
| theory which define properly what according to it a
| measurement is, an not take measurement and observations
| as axioms.
| eigenket wrote:
| Will you grant me that it is at least possible to derive
| Bell's inequality by listing out a complete set of
| assumptions (including assumptions that define what a
| measurement is and what observations are)?
|
| Of course you personally may disagree with some of these
| axioms (indeed, if you take Bell's theorem seriously you
| must), but certainly it is possible to list them, and
| thereby derive Bell's inequality?
| GistNoesis wrote:
| Bell's theorem is a theorem. If hypothesis applies
| conclusion must follow. That's math. Everything is fine
| with it (They are a reformulation of "Bonferroni
| inequalities" or "Boole's_inequality" by the way).
|
| You've got to reframe the problem so that Bell's theorem
| doesn't apply. When you build your theory, if you manage
| to define what a measurement is, so that you don't
| satisfy the hypothesis of the Bell's theorem, you get to
| avoid having to have its conclusions.
|
| One of Bell's theorem implied hypothesis is that
| measurements/observations are probabilities, so by
| defining measurement instead as a conditional
| probability, you get to avoid being subjected to Bell's
| inequalities.
|
| It's inductive reasoning, you don't get truth you only
| get self consistency, and a model that looks much nicer
| than QM.
| eigenket wrote:
| > You've got to reframe the problem so that Bell's
| theorem doesn't apply. When you build your theory, if you
| manage to define what a measurement is, so that you don't
| satisfy the hypothesis of the Bell's theorem, you get to
| avoid having to have its conclusions.
|
| This (in my opinion) a bad way of explaining how the
| standard reasoning goes. We start with a list of
| assumptions, we prove this inequality which it turns out
| is not satisfied, we reject (at least) one of our
| assumptions. These is no crackpottery here, this is the
| norm.
|
| > by defining measurement instead as a conditional
| probability
|
| This sounds like it probably doesn't get you anywhere,
| but I'll bite - what are we conditioning on? In the
| standard formulation of Bell's theorem they are
| conditional on the "hidden variable" we are assuming
| exists, as well as any relevant measurement settings but
| it sounds like you're imagining something wilder than
| that.
| GistNoesis wrote:
| >what are we conditioning on?
|
| The local hidden state, but you don't get to set it from
| inside the universe when you do an experiment (this local
| hidden state is unobservable).
|
| From inside the universe based on this hidden state,
| everything behave classically, pseudo-randomly based on
| the local hidden state.
|
| But because you don't get to set the local hidden state
| during your experiment if you want to calculate the
| probabilities, you have to integrate over the possible
| values of the unknown hidden state, and this allows you
| to recover the strange looking quantum correlations.
|
| Doing repeated experiment inside a universe mean picking
| a different initial local hidden state (because it's
| unobservable).
|
| [Spoiler ahead] The original idea is not from me, if you
| want the nitty gritty details, look at the work of Marian
| Kupczynski (Closing the Door on Quantum Nonlocality
| https://philarchive.org/archive/KUPCTDv1 ). Or his more
| recent works.
|
| I have made a straight forward implementation (3 years
| ago) of it to convince myself with a Monte Carlo
| simulation : https://gist.github.com/unrealwill/2a48ea092
| 6deac4011d268426... [End Spoiler]
| eigenket wrote:
| 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.
|
| Your numerical code is impossible for me to read without
| some basic idea of what you're trying to show, but I'd
| like to point out that numpy has functions like
| np.radians, and np.deg2rad to convert from degrees to
| radians, you don't have to make your own.
| DebtDeflation wrote:
| I commented awhile back on another thread that:
|
| I think, ultimately, there are only 3 possible
| explanations for the paradoxes of the quantum world. 1)
| superdeterminism (everything including our choices in
| quantum experiments today were fully determined at the
| instant of the Big Bang), 2) something "outside" our
| observable reality acting as a global hidden variable
| (whether something like the bulk in brane cosmology or
| whatever is running the simulation in simulation theory)
| or 3) emergent spacetime (if space and time are emergent
| phenomena then locality and causation are not
| fundamental).
|
| You seem to be suggesting something similar to option 2.
| Or am I misunderstanding?
| meroes wrote:
| MWI is not local according to all the big names I read; Lev
| Vaidman, Tim Maudlin, and Im pretty sure David Wallace too.
| eigenket wrote:
| You can probably define locality in a way that MWI is
| nonlocal, but you can also definitely define it in a way
| such that MWI is local.
|
| For me the most important thing about nonlocality is the
| lack of any "action at a distance", MWI satisfies this, but
| if you make more stringent demands it might not satisfy
| those.
| hnax wrote:
| This 'crackpot physicist' is still alive and kicking and,
| indeed, as per the (analytical) induction requirement to make
| the case, his work deserves a careful reading to assume the
| geometric algebraic understanding of QM (for the 'crackpot's
| latest, see: https://www.linkedin.com/posts/joy-christian-
| oxford_comment-...)
| Strilanc wrote:
| If you want to try your hand at violating Bell inequalities,
| there are widgets in [1] that allow you to input strategies (as
| javascript) for Alice and Bob. It continuously performs Monte
| Carlo sampling of the strategies and presents their success rate.
|
| There's a classical-only widget, that goes through quite some
| contortions behind the scenes to prevent cheating via writing to
| global variables, and a quantum-allowed widget where that kind of
| cheating is possible due to the underlying implementation
| cheating in precisely that using-globals way in order to
| correctly simulate the quantum mechanics.
|
| Anyways, I've had a few people tell me playing around with the
| widgets helped them understand the inequality.
|
| [1]: https://algassert.com/quantum/2015/10/11/Bell-Tests-vs-No-
| Co...
| tsimionescu wrote:
| > This is known as a Bell inequality. It captures the essential
| limitation imposed by any theory based on local hidden variables
| -- theories that adhere to classical notions of determinism (no
| random chance in the measurement apparatus), locality (no faster-
| than-light influences) and realism (pre-existing properties).
|
| Obligatory reminder that there is an extra assumption here: the
| assumption that the result of the coin flip is not correlated to
| the hidden state of the particle. If when receiving a particle in
| stage a_H your coin flip always leads to, say, HH, then you will
| break Bell's inequality even if all the other assumptions hold.
| Theories that have this property are called "superdeterministic".
| jmmcd wrote:
| This was really excellent - for many of us, code helps to make
| mechanisms concrete, and it forces every single thing to be
| pinned down, and not hand-waved away.
|
| (Like another commenter, I was also hoping for a
| direct/standalone explanation for why the red matrix is
| transposed.)
| westurner wrote:
| Hidden variable theory: https://en.wikipedia.org/wiki/Hidden-
| variable_theory
|
| 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 e [\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 e > 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 e>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 e>(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?
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