## Simple violation of Bell inequalities

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### Re: Simple violation of Bell inequalities

Yes, alternative "(A=B) OR (A=C) OR (B=C)" is always true (is tautology) for binary variables - what can be seen as example of Dirchlet's principle.

Or from probabilistic perspective, assume there is some probability distribution among 8 possibilities: sum_ABC pABC = 1
Pr(A=B) = p000 + p001 + p110 + p111
Pr(A=C) = p000 + p010 + p101 + p111
Pr(B=C) = p000 + p100 + p011 + p111
Pr(A=B) + Pr(A=C) + Pr(B=C) = 2p000 + 2p111 + sum_ABC pABC >= 1

Absolutely obvious, but violated e.g. by QM or MERW. No "local realism", but just standard probabilistics ( https://en.wikipedia.org/wiki/Probability_axioms ):
- first axiom: assumption of existence of probability distribution among all possibilities (that unknown = unmeasured), second axiom: Pr(Omega) = 1,
- third axiom - standard rule: "probability of alternative of disjoint events is sum of their probabilities" Pr(A=B) = p000 + p001 + p110 + p111

To violate it, we need to replace these two natural assumption with Born rule:
- instead of probabilities of events, there exist their amplitudes,
- Born rule: "probability of alternative of disjoint event is proportional to sum of squares of their probabilities": Pr_11 (A=B) ~ ((psi000 + psi001)^2 + (psi110 + psi111)^2)
Indeed formally there should be added indexes of which variables are measured - to distinguish just being unknown, from being unmeasured: the Born rule summation is over unmeasured variables.

Then we can violate inequalities derived using standard probability, like above.
for example assuming amplitude:
psi000 = psi111 = 0
psi001 = psi010 = psi011 = psi100 = psi101 = psi110 = 1
we get
Pr_11 (A=B) = ((psi000 + psi001)^2 + (psi110 + psi111)^2) /(sum_AB (psiAB0 + psiAB1)^2 ) = 1/5
Pr_11(A=B) + Pr_13(A=C) + Pr_23(B=C) = 3/5 violating the inequality.

Please write if I can elaborate on something.

### Re: Simple violation of Bell inequalities

Jarek wrote:Can you violate "tossing 3 coins, at least two are equal": Pr(A=B) + Pr(A=C) + Pr(B=C) >=1 inequality without Born rule/time symmetry?

Jarek, I just don't understand what you mean. If *three coins* are tossed, at least two are equal. The inequality Pr(A=B) + Pr(A=C) + Pr(B=C) >=1 follows from the fact that the union of the three events {A=B}, {A=C},{B=C} is certain.

You can't violate that inequality in the context in which it is valid. You are not talking about three coins being tossed. You are talking about three different sub-experiment in each of which only two coins are tossed. So you should have given "Pr" a subscript. There is Pr_11(A=B), Pr_13(A=C), Pr_23(B=C). The subscripts refer to which coins are tossed, and symbols A, B, C refer to the outcomes of the three coins which are being tossed, in different combinations, in different sub-experiments.

Your language is typical, I'm afraid, of the language of physicists. Your notation is inadequate. It is misleading. It does not make it easy to communicate with mathematicians or statisticians. Of course, most physicists feel no need whatsoever to communicate with mathematicians or statisticians. So it doesn't matter. But that is why there is so much *noise* in fifty years of physicists talking about Bell's theorem. They don't have an adequate language to talk about it, they are too arrogant to realise this, and hence they go round and round in circles.

### Re: Simple violation of Bell inequalities

Local-realistic perspective would choose some probability distribution for 2^3 = 8 possibilities for 3 binary variables, then summing probabilities of above 3 possibilities we would always get at least 1.
However, Born rule allows to get below: https://arxiv.org/abs/1212.5214

This is too simple situation for some hidden assumptions - you can convince people if you can violate that.
But it needs a short concrete construction - starting with explanation what measuring exactly 2 out of 3 means - how does it differ from just not knowing the third one?

### Re: Simple violation of Bell inequalities

***
Your question is irrelevant for the local-realistic underpinning of quantum correlations, which is what Bell's "theorem" is all about. Please read my papers I have linked.

***

### Re: Simple violation of Bell inequalities

Can you violate "tossing 3 coins, at least two are equal": Pr(A=B) + Pr(A=C) + Pr(B=C) >=1 inequality without Born rule/time symmetry?

Or derive some other basic consequence of Born rule, for example what is the stationary probability distribution for particle in infinite [0,1] well?
Standard diffusion says uniform: rho = 1.
QM says rho ~ sin^2 ... where does it come from?
Uniform ensemble of past paths says rho ~ sin
Uniform ensemble of complete paths says rho ~ sin^2

### Re: Simple violation of Bell inequalities

***
No retrocausality is needed to overcome Bell's so-called "theorem", as I pointed out way back in 2007: https://arxiv.org/abs/quant-ph/0703179.

For a comprehensive local-realistic framework without needing retrocausality, see my latest Royal Society paper: https://royalsocietypublishing.org/doi/ ... sos.180526.

***

### Re: Simple violation of Bell inequalities

https://arxiv.org/pdf/1906.04313
K.B. Wharton, N. Argaman, Bell's Theorem and Spacetime-Based Reformulations of Quantum Mechanics

In this critical review of Bell's Theorem, its implications for reformulations of quantum theory are considered. The assumptions of the theorem are set out explicitly, within a framework of mathematical models with well-defined inputs and outputs. Attention is drawn to the assumption that the mathematical quantities associated with a certain time and place can depend on past model inputs (such as preparation settings) but not on future inputs (such as measurement settings at later times). Keeping this time-asymmetric assumption leads to a substantial tension between quantum mechanics and relativity. Relaxing it, as should be considered for such no-go theorems, opens a category of Future-Input Dependent (FID) models, for which this tension need not occur. This option (often called `retrocausal') has been repeatedly pointed out in the literature, but the exploration of explicit FID models capable of describing specific entanglement phenomena has begun only in the past decade. A brief survey of such models is included here. Unlike conventional quantum models, the FID model parameters needed to specify the state of a system do not grow exponentially with the number of entangled particles. The promise of generalizing FID models into a Lorentz-covariant account of all quantum phenomena is identified as a grand challenge.

### Re: Simple violation of Bell inequalities

Thanks, interesting paper.
General relativity is example of Lagrangian mechanics, Einstein's equation is time-symmetric, can be seen as equilibrium condition for spacetime as "4D jello" - Einstein's block universe.
And it doesn't really emphasize time direction - only chooses this direction locally, e.g. switching with space direction below black hole horizon ... this choice of time direction is not even objective: changes with velocity of observer through Lorentz transform.

While Lagrangian mechanics has built in philosophy of time/causality in its mathematics (a matter of extracting and accepting it), QM leaves a lot of freedom for choosing own philosophy - interpretation.
To combine them, it is crucial to choose a QM philosophy which is compatible with GR.

Time evolution way of thinking: "psi(t)" emphasizes time direction, would require modification e.g. with Lorentz transform as it changes time direction, becomes something really strange below black hole horizon ...
So can we loosen this time dependence for compatibility with GR - treat time as just a local direction in 4D spacetime?
Yes we can - for example path ensembles, or more generally ensembles of Feynman's graphs/scenarios in perturbative QFT, are compatible with GR - just a matter of choosing and evaluating such paths/scenarios using GR, for example only inside light cones, e.g. rotating 90deg below black hole horizon.

So path/scenario ensemble seems a safe philosophy/interpretation of QM.
Propagating such paths from -infinity and +infinity to this moment, we get TSVF: https://en.wikipedia.org/wiki/Two-state ... _formalism
Considering the simplest: uniform path ensemble (MERW), we analogously get Born rules from time symmetry: one psi from past, second from future.
Having construction for Born rules, we can construct Bell violation examples.

### Re: Simple violation of Bell inequalities

I agree entirely that we should modify our philosophical (and actually, our deep neuro-psychological) preconceptions in the light of physics and successful mathematical description of physical reality! I think that QM leads us to deny one of Aristotle’s philosophical prejudicies.

Before I do that reading, which you recommend: you could maybe take a look at

https://doi.org/10.1038/s41467-019-11579-x
Bell’s theorem for temporal order
Magdalena Zych, Fabio Costa, Igor Pikovski & Časlav Brukner
Time has a fundamentally different character in quantum mechanics and in general relativity. In quantum theory events unfold in a fixed order while in general relativity temporal order is influenced by the distribution of matter. When matter requires a quantum description, temporal order is expected to become non-classical—a scenario beyond the scope of current theories. Here we provide a direct description of such a scenario. We consider a thought experiment with a massive body in a spatial superposition and show how it leads to entanglement of temporal orders between time-like events. This entanglement enables accomplishing a task, violation of a Bell inequality, that is impossible under local classical temporal order; it means that temporal order cannot be described by any pre-defined local variables. A classical notion of a causal structure is therefore untenable in any framework compatible with the basic principles of quantum mechanics and classical general relativity.

### Re: Simple violation of Bell inequalities

Just please take a closer look at the simplest time-symmetric model (MERW): just uniform ensemble of paths on a graph, or equivalently: random walk chosen accordingly to the maximum entropy principle.
This is just simple combinatorics/probability theory, already explaining how e.g.:
- diffusion on [0,1] instead of "obvious" uniform stationary density, can lead to "quantum" rho ~ sin^2 localization: the number of paths is reduced near a boundary,
- where the squares of Born rules come from: directly from time symmetry. Having Born rule you can violate Bell-like inequalities.
My PhD thesis about MERW: https://fais.uj.edu.pl/documents/41628/ ... 74256fd065

And generally instead of enforcing made up human philosophies on physics ... if we want to really understand it, we should go the opposite way: look at what describes physics well - successful quantitative models like extremely universal Lagrangian mechanics, and translate their assumptions into philosophy we use to interpret physics.

Humbly extract philosophy from physics, instead of arrogantly trying to enforce own human philosophy on it.

### Re: Simple violation of Bell inequalities

Thanks! I am going to study this in more detail. Also, consult with my colleagues working in interacting particle systems. It's big business in mathematics in the Netherlands.

### Re: Simple violation of Bell inequalities

Dear Richard,

I accept realism, but point misunderstanding in standard view on localism: enforcing time asymmetry. Replacing it with the proper: time-symmetric localism, like in Einstein equation (generally Lagrangian mechanics) or path ensembles, the problems disappear.
For example here is derivation of Born rules from uniform path ensemble on a graph with adjacency matrix A:
https://en.wikipedia.org/wiki/Maximal_e ... derivation

The sequential Stern-Gerlach diagram is from Wikipedia article, tested experimentally.
To take it to path ensembles, we need to extract the measurement rule from it - for my Bell violation construction I used rule: "during measurement, its outcome cannot change" - e.g. spin stays in once chosen alignment in magnetic field.

Regarding universe, this is also not mine but a general view e.g. in Zurek's einselection ( https://en.wikipedia.org/wiki/Einselection ): wavefunction collapse is from interaction with environment (e.g. in S-G: spin interacts with external magnetic field).
Universe does no longer have external environment to interact with, hence wavefunction of universe has only unitary evolution (deterministic, time symmetric).

I don't have base to claim anything regarding anthropic principle, but generally (after working on soliton particle models) don't think our type of physics is that unlikely.
What is nontrivial is that proton is lighter than neutron - required to have atoms, naively should be the opposite as "charge weights" ... but having that, there are atoms, which can form complex structures ... maybe life.

I am just supporter of Lagrangian mechanics - mathematics we successfully use in all scales, e.g. QFT, EM, GRT - which complete acceptance is nontrivial due to its time/CPT-symmetry.
It is deterministic, allowing for three mathematically equivalent perspectives: the most useful to understand e.g. Born rule is time-symmetric: https://en.wikipedia.org/wiki/Principle_of_least_action
Wanting to use equivalent forward in time evolution (Euler-Lagrange equation) requires this pre-determinism mind acrobatics of initial state prepared for all future measurement ... but time-symmetric perspective does not need this acrobatics, is better for understanding.

### Re: Simple violation of Bell inequalities

Jarek wrote:Richard, I only reject standard time-asymmetric "local realism", replacing it with time-symmetric (as in the rest of physics) "4D local realism" like path ensembles or in Einstein's equation - what seems to repair all the problems, e.g. also leads to Born rules and resulting Bell violation.
...
Please specify where do you see a problem here and I will gladly explain.

Dear Jarek

I understand from your reply that your definitions of "local" and "realism" differ from those used by most of those working in quantum foundations. Indeed it is easy to restore "local realism" to quantum mechanics: redefine "local" and redefine "realism". I am impressed that you are able to recover the Born law and the quantum correlations in your framework. Excellent. It is well worth studying in more detail.

You say you can easily explain a sequence of Stern Gerlach measurements in your framework. But there is no problem at all in finding a local and realistic description of what goes on in a sequence of Stern Gerlach measurements, anyway.

I would like to know how you would use your framework to give a local realistic description of the life of the whole universe. There is no outside observer. There is just one closed system. Cosmologists are well able to make computer simulations which occasionally generate universes much like ours. But more often they generate universes which are very different. Either they live very short spectacular hot lives but without any generation of interesting structures; or they live very, very long and completely boring cold lives in which nothing ever happens at all.

The anthropological principle may be thought to "explain away" this problematic feature - universes unlike ours would not actually produce environments like ours - environments able to support the evolution of effectively sentient beings which are able to ask the questions "where do we come from, where are we going?".

So you are a supporter of pre-determinism? The evolution of the universe is completely deterministic, violation of Bell inequalities is explained because the measurement settings which the experimenter thought she was freely choosing, were already fixed at the time of the big ban? The principle of information causality seems to be a fundamental principle of physics and it enforces the Tsirelson bound - so Bell-CHSH can be violated, quantum non-locality is for real, but there is no "action at a distance".

### Re: Simple violation of Bell inequalities

Richard, I only reject standard time-asymmetric "local realism", replacing it with time-symmetric (as in the rest of physics) "4D local realism" like path ensembles or in Einstein's equation - what seems to repair all the problems, e.g. also leads to Born rules and resulting Bell violation.

Applying Jaynes maximal entropy principle is the difference between standard random walk/diffusion and MERW - the former usually only approximates it, while MERW is the unique random walk in agreement with this principle - the safest choice if having no additional information.
Lots of papers about it: https://scholar.google.pl/scholar?q=max ... ndom+walks

This looking subtle difference leads to huge consequences, for example imaging random walk on [0,1] range:
- standard random walk says that stationary distribution is rho = 1
- MERW and QM says that stationary distribution is rho ~ sin^2(x)

Why maximal entropy choice is the safest one?
This is mathematically universal law as entropy ~ lg(number of possibilities).
Hence focusing on possibilities defined by parameter p, maximizing it means focusing on possibilities which are asymptotically dominant.
For example knowing only that "we have n white and black balls", number of possibilities for p percentage of white is binomial(n, pn) ~ exp(n * h(p)), the p=1/2 possibilities asymptotically (n -> infinity) dominates: it is the safest choice if knowing nothing more.

Regarding the measurement "problem", while entanglement of macroscopic object like cat is doubtful (is really tough to distinguish from just not knowing), maybe let's focus on microscopic measurement.
Stern-Gerlach ([url=https://en.wikipedia.org/wiki/Stern–Gerlach_experiment]en.wikipedia.org/wiki/Stern–Gerlach_experiment[/url]) is great idealization of microscopic measurement, allowing also for sequential measurements:

And Stern-Gerlach can be understood: parallel and anti-parallel magnetic dipole alignments are local energy minima in strong magnetic field, so one of them has to be chosen.

Please specify where do you see a problem here and I will gladly explain.

### Re: Simple violation of Bell inequalities

Forget local realism. We have both rejected it. It is not *our* mutual problem.

Our *mutual problem* is whether or not there is a measurement problem in quantum physics. Is the Schrödinger cat paradox *merely* a paradox (an *apparent* contradiction) or is it a *real* contradiction which needs to be resolved by changing something in the fundamental theories which we use?

Plenty of more renowned persons than me think that it is a real problem. The literature is vast. You seem to be oblivious of it. Of course, I am also oblivious of huge portions of mathematical physics, not to speak of my larger ignorance of the larger field of "theoretical physics".

Anyway, please comment on the Schrödinger Cat Paradox and in particular, on the list of possible approaches to the "paradox" at https://en.wikipedia.org/wiki/Schr%C3%B6dinger's_cat.

Where do you place yourself in this large menu of "resolutions" of the "problem"?

Or do you adopt the common physicists' "shut up and calculate" point of view, also known as "the FAPP trap" (FAPP = for all practical points of view); terminology introduced by John S. Bell.

BTW, I don't find Jaynes' maximum entropy principle in any way compelling. I don't understand why some people become such fanatical adherents of the principle. It seems to belong to a subjective view of what probability means which I don't find very useful as a basis for real physics.

Jaynes was an original thinker who made many important contributions to science, but he was also dogmatic and he did make big mistakes, too. He is not one of my greatest scientific heroes. So recruiting Jaynes to support your point of view does not bear any weight with me, on the contrary, it weakens your argument.

### Re: Simple violation of Bell inequalities

After "you do not respond to my comments" unsupported criticism, again only vague general remarks, zero meritorious arguments, responses to questions regarding your statesments ...

I have tried reading Belavkin paper, but didn't find it resolve the central problem here: of local realism. Asking for elaboration, you have only brought "angelic forces" - type of "explanation" I am not interested in.
And I don't know how you understand this theory - please specify your point based on it and I would gladly address your meritorious remarks. Like while trying to understand what do you mean in "past are particles, future are fields" claim. Unfortunately you don't support your statements.

And there are no races, but rather stagnation for a century - due to stubbornness on time asymmetry of human intuition bias - even after being proven wrong e.g. by Bell violation, and successfully using time/CPT symmetric models in all scales.

### Re: Simple violation of Bell inequalities

Jarek wrote:Just be a mathematician - replace human bias with acceptance of mathematics.

Dear Jarek, I am a mathematician, I do accept your mathematics, I hope to have time to assimilate it, but I do have some other priorities. Mirrorwise, I referred you to the impeccable but difficult mathematics of Slava Belavkin's "eventum mechanics". You refused to pay it any attention at all because you are not prepared to learn some basic mathematical notions. You weren't even prepared to take a look at some simple tutorial introductions. We have a standoff - an obstinate mathematician facing an obstinate physicist! But it doesn't matter. We're not the only exemplars of our two races in the world. Hopefully, you can "sell" your approach to other people. I have the same hope regarding mine; eventually, the two races will meet and breed.

### Re: Simple violation of Bell inequalities

I am only saying that if there would be some faster-than-light causality, we should be able to do it in a controllable way - sending information ... but we cannot - so this way of thinking is incorrect, this option is invalid.

I am a mathematician and my specialism is not even mathematical physics, but mathematical statistics and probability theory.

So be one - instead of enforcing human time-asymmetric intuition bias, which was shown to be wrong ... just look at mathematics of accepted theories we successfully use from QFT to GR: with time/CPT symmetry at their hearts.
Don't make up "particles in past, fields in future"-like hypotheses you cannot defend, but just accept mathematics.
E.g. in general relativity there is nothing special in time - just one of 4 directions of spacetime, chosen locally, e.g. switching with space below black hole horizon ...

To understand statistical properties, the safest assumption is (Jaynes) maximal entropy principle: https://en.wikipedia.org/wiki/Principle ... um_entropy - for example it is at heart of statistical physics in its uniform/Boltzmann distributions.
To include time-symmetry in such models, we need to do statistical physics for time-symmetric objects, like paths - the simplest such model is uniform path ensemble - MERW.

This is exactly your field: statistics and probability theory, but I haven't seen any meritorious arguments about it - only some vague criticism like "Yet you have no solution for the measurement problem" - without any clarification, elaboration, even after asking multiple times and waiting for a month.
you do not respond to my comments

I am trying to address all your comments, please just repeat if I miss something crucial. Which do you think I have missed?
It seems that you do not realise that the Schrödinger cat paradox is an unsolved problem for creating a coherent mathematical model of quantum mechanics

Please specify where do you see a problem here?
Transferability of quantum properties to macroscopic physics means they are more universal - that the search for classical-quantum boundary is doomed to failure.
Example of universal theory is statistical physics, like uniform path ensemble (MERW), e.g. predicting rho ~ sin^2 stationary density even for macroscopic [0,1] potential well ... but we don't observe such "quantum" localization e.g. in gas tanks - having rather rho ~ const density.
Of course, if we accept predeterminism, then we do have a coherent picture. But to my taste, not an attractive one.

To make it attractive, it again requires replacing human intuition bias with mathematics: Lagrangian mechanics has 3 mathematically equivalent formulations, predeterminism is for forward in time E-L evolution.
Using equivalent (time-symmetric) action optimization formulation/perspective ( https://en.wikipedia.org/wiki/Principle_of_least_action ): history is optimized for conditions in the past and in the future, we get explanation why initial state of our Universe was already prepared for all future measurements.

Just be mathematician - replace human bias with acceptance of mathematics.

### Re: Simple violation of Bell inequalities

Jarek wrote:Richard, MERW is not a formulation of quantum theory, only the simplest path ensemble model (uniform) - still getting some crucial "quantum" properties - with simple explanations.
Alternatively, MERW is just random walk done right: accordingly to (Jaynes) maximal entropy principle, this way repairing disagreements of standard diffusion models - usually only approximating this crucial principle.
MERW misses quantum interference (still having Anderson localization), in contrast to QM its excited states are not stable - deexcitate to the ground state without need of interaction with environment.

In my opinion, quantum jumps are for real and they are indeed in some sense non-local. So I keep realism (objectivity) and I endorse "quantum non-locality". Which is *not* action at a distance. QM does not allow action at a distance. The non-locality is of a rather subtle kind since it is made possible by irreducible randomness.

Where do you see allowance for jumps in theories we use? Why we cannot exploit them e.g. to send information faster than light?
In Lagrangian mechanics such discontinuities would mean infinite energy (~derivative^2).
Even QFT does not allow for them - e.g. building physics with ensembles of Feynman diagrams: scenarios with continuous trajectories.

Instead of jumps/discontinuities, theories we successfully use in all scales have time/CPT symmetry - they have locality, but different: symmetric one, in contrast to the one in "local realism".

To summarize, there are basically two ways for repairing "local realism":
1) just accept time/CPT symmetry of physics: replace asymmetric locality with symmetric one, or
2) add jumps/discontinuities against all physics we use ... but if they are true, so why we cannot use them, e.g. to send information faster than light with such jumps?

Your last question has been answered by recent work on "Information Causality". Are you aware of it? It answers your question, with beautiful theory and ongoing research. There is a very famous paper in "Nature". https://en.wikipedia.org/wiki/Information_causality

I'm sorry I cannot respond to the rest of your remarks. I am not a physicist! I am a mathematician and my specialism is not even mathematical physics, but mathematical statistics and probability theory. Similarly, you do not respond to my comments. It seems that you do not realise that the Schrödinger cat paradox is an unsolved problem for creating a coherent mathematical model of quantum mechanics [at least, according to very many people working in quantum foundations, ie in logic, philosophy and meta-physics]. Of course, if we accept predeterminism, then we do have a coherent picture. But to my taste, not an attractive one. I think that QM has been telling us for 100 years that cherished physical principles (the ones you keep paying homage to) need to be reconsidered. Still physicists are not ready to move on...

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