Most arguments against any hidden variable theories in quantum mechanics are based on Bell's theorem and consequences. In the proof of Bell's theorem, there is a hidden assumption related to the countable additivity of the Lebesgue measure. If we look at two entangled quantum systems as synchronized chaotic systems, then it is not at all obvious that the Lebesgue measure has to be countably additive, specially on a chaotic attractor. Therefore, the hidden assumption in Bell's theorem is not valid, the proof of the theorem fails in this case, and so do many arguments against hidden variable theories. As for the GHZ entanglement, and associated paradox (involving three quantum systems), if you carefully look at how three chaotic systems can be synchronized (various types of synchronization), I am certain that the paradox dissipates.

There have been tomes of papers and books written about quantum nonlocality, and it is all basically based on Bell's theorem and the GHZ paradox. Could Einstein's "spooky action at a distance" be explained in terms of chaos theory? Could it all be explained based on synchronized chaotic systems?

  • $\begingroup$ Do you have a reference where these ideas come from? $\endgroup$ – Bruce Greetham Nov 13 '16 at 3:36
  • $\begingroup$ I don't know the answer to this, but found some surprising results on studies of the similarities. Specifically I saw that there are papaers that cite that there are classical chaotic systems that violate Bell's inequalities, which supposedly only quantum systems could violate, and was the result that led to Bell inequality experiments being seen as a proof that there are no local hidden variables, i.e., that quantum mechanics uncertainty is definitive. Has that been disproven? I'll re-find the ref, but it's not hard to google the topic of chaos sync and entanglement, my iPad is acting up $\endgroup$ – Bob Bee Nov 13 '16 at 7:52
  • $\begingroup$ Here is one such paper folk.uib.no/gdu041/fplprf2.pdf $\endgroup$ – Bob Bee Nov 13 '16 at 7:57
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    $\begingroup$ So you think you knew the answer all along and the question is a Trojan horse. There are plenty experiments confirming the Bell etc inequalities for quantum theory are not possible with classical theory. I read Bohm as a junior in college and promptly wrote it off. That one papaer I found may be confirmed or not, I doubt it since I've not seen mainstream publications on the validity of hidden variables in any form. If you have some validated results published in mainstream publications that classical chaos violate the Bell inequalities then give us a reference. No need to answer otherwise. $\endgroup$ – Bob Bee Nov 14 '16 at 0:22
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    $\begingroup$ What is the connection between Bohm's theory and chaos? $\endgroup$ – user154997 Jul 19 '17 at 1:21

In the proof of Bell's theorem, there is a hidden assumption related to the countable additivity of the Lebesgue measure.

Let's see whether this statements holds for the CHSH version of Bell's inequalities [2], a variant of which is universally used for all experimental tests.

We have a pair of particles, labelled 1 and 2 in the following, and two detectors $\newcommand{\DA}{\mathcal{A}}\newcommand{\DB}{\mathcal{B}}\DA$ and $\DB$: particle 1 (resp. 2) will interact with detector $\DA$ (resp. $\DB$). Moreover detector $\DA$ (resp. $\DB$) has a tunable setting $a$ (resp. $b$). About the state $\lambda\in\Lambda$ of the pair, we only know a probability measure $\rho$ on $\Lambda$. The key hypothesis is then locality: the measurement by $\DA$ (resp. by $\DB$) depends only on $\lambda$ and $a$ (resp. $b$), i.e. there are functions $A(a,\lambda)$ and $B(b, \lambda)$. We restrict ourselves to the case where the outcome of the measurement can only be $\pm 1$ because this is the relevant case experimentally. The correlation between the measures is then

$$P(a,b)=\int_\Lambda A(a,\lambda)B(b,\lambda)d\rho(\lambda).$$

Then let's prove the CHSH inequality

$$R(a, a', b, b') = |P(a',b) - P(a,b) + P(a',b') + P(a,b')| \le 2.$$


$$|R(a, a', b, b')| \le \int_\Lambda \left|(A(a', \lambda) - A(a, \lambda)) B(b, \lambda) + (A(a', \lambda) + A(a, \lambda)) B(b', \lambda) \right|d\rho(\lambda).$$

Since $A$ can only take the values $\pm 1$, either $A(a',\lambda)$, and $A(a,\lambda)$ have the same value, then the integrand is $\pm 2 B(b', \lambda)$, or they have different values, and the integrand is $\pm 2B(b,\lambda)$. Since $|B|\le 1$, the inequality follows.

Now let's review your claim. I fail to see where countable additivity is a hidden assumption in this demonstration. We only had to assume that $\lambda\mapsto A(a,\lambda)$ and $\lambda\mapsto B(b,\lambda)$ were measurable. To simplify the discussion, I'll assume that $\Lambda \in \mathbb{R}^n$ and that $\rho$ is the Lebesgue measure. The issue I have then is that whether all functions are measurable or not depends on the set theory one chooses to use as foundation of mathematics.

  • There are models of ZF + DC in which one can define a $\sigma-$additive Lebesgue measure such that all subsets of $ \mathbb{R}^n$ are measurable. [1, Appendix: Models] Here DC stands for the Principle of Dependent Choices: for every pair $(X,\rho)$, where $X$ is a non-empty set and $\rho$ is a relation on $X$ such that for every $x\in X$, there exists $y\in X$ with $x\rho y$, there exists a sequence $(x_n)$ in $X$ with $x_n\rho x_{n+1}$ for each $n\in\mathbb{N}$.
  • In ZFC, the Axiom of Choice can be used to construct non-measurable sets of $\mathbb{R}^n$. [1, Theorem 5.5]

Afaik, which set theory is used has never had any impact on physics. That does not logically invalidate your argument but it means, at least, that if you were to need to rely on ZFC to make your case, your claim would be extraordinary enough to warrant extraordinary evidences.

[1] Horst Herrlich, Axiom of Choice, 2006, Springer [2] Clauser, Horn, Shimony, and Holt, Phys. Rev. Lett. 23 (1969) 880

  • $\begingroup$ I added an answer to my own question, because the comments do not have enough space. I clarified some things there. We have to work within ZFC, otherwise, we cannot even talk about the Lebesque measure. $\endgroup$ – Cristian Dumitrescu Jul 22 '17 at 16:27
  • $\begingroup$ If you read my last answer, your feedback will be aapreciated. $\endgroup$ – Cristian Dumitrescu Jul 22 '17 at 18:50
  • $\begingroup$ I edited my answer to make it crystal clear what I meant, and this addresses your claim we need ZFC. See also the caveat I added. I'll study the rest of your answer at a later time, sorry! $\endgroup$ – user154997 Jul 26 '17 at 9:25

I have only a basic knowledge about Bell's theorem, hidden variables and Lebesgue measure. However, I think, it is more or less obvious that the answer to your question is "No".

Chaotic motion is only a one type of behavior that dynamical systems can exhibit. There exist motions like limit cycle or intermittency in various dynamical systems. Now, when we talk about any motion in case of physical system modeled as a dynamical system, implicitly it is always assumed that the variables of the dynamical system are "macrovariables" meaning that they correspond to the description in the thermodynamic limit. This means that, you can't talk about a dynamical system (deterministic or stochastic) consisting of a single atom or electron. This remains true even when you go to "quantum chaos" where the description is still in the form of a classical macroscopic variables and the Hamiltonian of the system is then treated quantum mechanically.

It is then clear that when one talks about synchronization, they always mean that macroscopic variables are synchronized. So if the synchronization happens in say two physical oscillators, the individual atoms in them would still be in completely different states and only the average of the motions would be same. All in all, I don't see even a distant connection between entanglement and the synchronization in dynamical systems. Hope this helps.

  • $\begingroup$ Your argument is correct in the context of most interpretations of the quantum mechanical mathematical apparatus, except from David Bohm's pilot wave model. In this model, you can precisely define what I mean by synchronized chaotic systems. $\endgroup$ – Cristian Dumitrescu Nov 13 '16 at 8:58
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    $\begingroup$ @CristianDumitrescu: True but only if the pilot wave theory is actually empirically true. It has already been reported to be wrong experimentally. $\endgroup$ – Peaceful Nov 13 '16 at 16:28
  • $\begingroup$ You are probably referring to Leggett inequalities. His result relies on the principle of noninvasive measurability, which I don't think it holds. Measurement (which means macroscopic effects ) is also related to chaotic evolution, if the macroscopic apparatus is included in the model, more precisely, connected to basins of attraction. All these 'paradoxes ' have the same origin. You cannot simulate the evolution of a chaotic system using an algorithm, or a universal Turing machine, working with finite precision. $\endgroup$ – Cristian Dumitrescu Nov 14 '16 at 6:01
  • $\begingroup$ The apparent randomness in quantum mechanics is an emerging type of pseudo randomness, the governing laws of which we can only approximate. $\endgroup$ – Cristian Dumitrescu Nov 14 '16 at 6:28
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    $\begingroup$ @CristianDumitrescu : Are you sure that you have question? $\endgroup$ – Peaceful Nov 14 '16 at 14:24

After some thought, considering the recent feedback to my question, I decided to clarify a few points in an answer, since the comments do not give me enough space.

Without a countable version of the axiom of choice, it is impossible to even define the Lebesque measure. We have to work within ZFC. Now, within ZFC, we can construct Lebesque non measurable sets, the Vitali sets (and they have certain fractal characteristics).

Let’s consider a toy model. This toy model does not have a direct connection to quantum mechanics, , even if some of the terminology is borrowed from quantum mechanics. We will first consider the double well oscillator:

x'= y
y'= x - x^3

In the phase space, this system has a saddle point at (0,0) , and centers at (1,0) and (-1,0) . Each of the neutrally stable centers is surrounded by a family of small closed orbits. There are also large closed orbits that encircle all three fixed points.

In our toy model, we will say that our system is in a superposition of two states, each one associated to a stable center.

We will now consider the periodically forced double well oscillator:

x'= y
y'= x - x^3 - δ·y + F ·cos(ω·t)

With δ=0.25 ,F=0.25 ,and ω=1.

For these parameter values, the system has two periodic attractors, corresponding to forced oscillations confined to the left or the right well. The basins of attraction have a complicated shape, and the boundary between them is fractal. Near the boundary, slight variation of initial conditions can lead to totally different outcomes.

In our toy model, we will call measurement, the dynamics of the periodically forced double well, for each of the two entangled systems, but we have to define entanglement.

In our toy model, entanglement corresponds to the fact that the process of measurement (the dynamics of the system represented by the periodically forced double well) starts with the same initial conditions, for each of the two entangled systems.

Under these definitions of entanglement and measurement, we see that when the first of the entangled systems ends up in the left stable well, the second system will also end up in the left stable well, and similar for the right stable well.

In order to study a phenomenon similar to the one studied by Bell (in the context of our toy model), we need to define different types of measurements, by simply varying the parameters δ, F, and ω , and the way the basins of attraction change (this is a complex problem, but it can be approached).

In the context of our toy model, we cannot follow the proof of Bell’s theorem, because of the fractal character of the domains involved in the phase space of the systems (and the connection to Vitali - type non measurable sets). That means that in our toy model, in principle, we could have correlations that go beyond classical results.

I think that something similar is happening in quantum mechanics, and we did not even touch the phenomenon of synchronized chaos yet. This toy model is only meant to prove a point of principle, and give some intuitive meaning to the issues discussed here.

Hidden variable theories can explain the strange phenomenon of quantum entanglement, without any reference to quantum non locality, which I don't think is a real phenomenon. The proof of Bell's theorem fails under certain circumstances, but nonlinearity is essential. The GHZ results can also be explained with this methodology.


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