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Bell's theorem is often interpreted to have destroyed local realism. This means something like the properties of the particle do not exist until they are being measured. I am not sure what does this exactly mean. Can there still be an algorithm to predict those measurements? Can anyone give me an example of a toy theory that is local but non-realist? How such "unreal" variable would be computed and what would the difference with local hidden variables be.

I am not asking for a quantum mechanical theory, just a local no real theory, how is a prediction made? is there an algorithm at all? if there is no algorithm, then what is it, a noncomputable law? a random law? what other options am I missing?

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    $\begingroup$ @brucesmitherson I'm not sure what algorithms have to do with it, but QM is literally where the concept of "non-realism" came from. It is non-realist in the sense that measurement results cannot be attributed to/determined by some hidden variable existing before the measurement itself. That's the typical way to express QM anyway, as a local non-realist theory. If you use a different formalism such as bohmian mechanics you can equivalently describe it as realist but non-local. Both choices are equivalent as far as I'm aware $\endgroup$ – glS Dec 6 '20 at 18:31
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    $\begingroup$ @brucesmitherson no, as I said above, these are different concepts. Bohmian mechanics is non-local but realist (it is fully deterministic, but admits action-at-a-distance). Standard QM is local (no instantaneous causal influences) but non-realist (no hidden variable explanations). See the answer below $\endgroup$ – glS Dec 6 '20 at 18:41
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    $\begingroup$ I think Tim Maudlin argues there is no such thing as "local nonrealism," so you may appreciate reading What Bell did. There is a reply article that gives counterarguments and argues for how there can be "local nonrealist" theories. At the least, I suspect the reply article would provide you with an understanding of what people are trying to convey. $\endgroup$ – Maximal Ideal Dec 6 '20 at 22:52
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    $\begingroup$ @ReasonMeThis Indeed, I should have been more careful with the terminology. QM produces nonlocal correlations. Nevertheless, it is local, in the sense of the theory being no-signaling. I'd say in general there is a difference between saying "QM is nonlocal" (which is generally taken to mean it allows for instantaneous causal influences) and saying "QM allows for nonlocal correlations (assuming a non-FTL description)", which is e.g. proven via Bell, CHSH etc. See sections (II.A.1) and (II.A.2) in the linked paper. QM is local, but predicts nonlocal correlations. $\endgroup$ – glS Dec 11 '20 at 8:43
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    $\begingroup$ @ReasonMeThis look, that's all just semantics. Until you precisely define your words you can say one thing or the other. The way I see it, the only meaningful statement is that QM cannot be described by a theory allowing only for correlations writable as $p(ab|xy)=\sum_\lambda p_\lambda p_\lambda(a|x)p_\lambda(b|y)$. The rest is chatter $\endgroup$ – glS Dec 12 '20 at 11:00
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An explanation is local if the behavior in one part doesn't depend on the actions in the other part. For example, local operations should commute if they are spacelike separated. You should be able to create two (classical) computer programs that simulate the system, and they don't need to communicate the actions they are taking to each other to produce correct output.

An explanation is realist if the state of the whole system can be described by some probability distribution. You should be able to reproduce the behavior of a local realist system by tallying up how the local computer programs you wrote from before respond to each of the probabilistic cases from the distribution.

An example of a non-realist system is... instead of the various cases from the probability distribution being independent, and adding up the usual linear way, they interact with each other in complicated ways. For example, you might see the system perform behavior like "if the chance that X happened is greater than 10%, cut the chance that Y happened in half" (which is not a valid probabilistic operation because it isn't linear; there is no way to turn it into a stochastic matrix with X and Y as basis vectors).

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  • $\begingroup$ My interpretation is that you mean that non-realist is equivalent to non markovian? Am I correct? $\endgroup$ – bruce smitherson Dec 7 '20 at 17:15
  • $\begingroup$ I think there might be a couple different definitions of markovian, but yes basically I mean it can be modelled by a state machine with probabilistic transitions. I don't know if that's the generally accepted definition but that's my reading of realism. $\endgroup$ – Craig Gidney Dec 7 '20 at 17:27
  • $\begingroup$ What would make a nonmarkovian process help reproduce the nonclassical correlations of QM as opposed to a markovian process? $\endgroup$ – bruce smitherson Dec 7 '20 at 17:35
  • $\begingroup$ @brucesmitherson I think you need the locality constraint or else you can do pretty much anything with a Markov process plus some hidden information. It's the combination of the two that's limiting. $\endgroup$ – Craig Gidney Dec 7 '20 at 17:41
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    $\begingroup$ @brucesmitherson Yes. For example, quantum mechanics is a local non-markov process that reproduces the QM correlations. (QM is local in the sense that spatially separated operators commute.) More generally, the space of non-markov processes include some effects that allow faster than light signalling so the locality constraint doesn't really mean anything if you're allowing non-markov. Post-selection is an example of a non-markov process that can be used for retroactive FTL signalling. $\endgroup$ – Craig Gidney Dec 7 '20 at 17:57
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The simplest and most used example would be, within quantum mechanics, the standard scenario of Bell's inequalities (in the CHSH form, for simplicity).

You have two boxes. Each one has two buttons, and according to some internal rule, it can respond to pressing a button with one of two outcomes (think two led lights on top of each box turning on or off). Denote the probability of observing the outcomes $a$ and $b$ when you press the buttons $x$ and $y$ with $p(ab|xy)$.

You can prove that there are probability distributions $p(ab|xy)$ which cannot be described by any kind of local, realist theory. This means that there is no hidden variable $\lambda$ with distribution $p(\lambda)$ such that $$p(ab|xy) = \sum_\lambda p_\lambda p_\lambda(a|x) p_\lambda(b|y),\tag1$$ where $p_\lambda(a|x)$ is the the probability of observing the outcome $a$ on the first box when you pressed the input $x$ and the hidden variable is $\lambda$.

One can prove that an entangled quantum state such as $|00\rangle+|11\rangle$, when using appropriate measurement bases, results in a probability distribution which cannot be written as (1). This proves that to describe measurement results in QM you need a theory that is either non-local or non-realist. See e.g. here for the proof.

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  • $\begingroup$ Thus, to you non-realist means no-hidden variables. Do you have any idea why a nonhidden variable model might result in nonclassical correlations? what advantage would a lack of hidden variables would give? $\endgroup$ – bruce smitherson Dec 7 '20 at 17:42
  • $\begingroup$ @brucesmitherson I'm not sure what you mean. The lack of a nonhidden variable model is almost the definition of "nonclassical correlation". It's nonclassical because you cannot reproduce it classically. These is all standard terminology by the way, see e.g. arxiv.org/abs/1303.2849 $\endgroup$ – glS Dec 10 '20 at 8:29
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I am not sure what you mean by "unrealism", unless it is the philosophy (adopted by many physicists) that the reality underlying quantum mechanics cannot be described, and that we should not even try to describe it. In this case one accepts the algorithm that probabilities for the results of measurement are given by the calculations of quantum mechanics (sometimes known as "shut up and calculate").

However, one should distinguish between realism and naive realism. Bell's theorem does not actually refute realism (local or otherwise) but it does refute naive realism, which I think is what you mean by "local realism". Indeed naive realism is already refuted by the mathematical structure of quantum mechanics even without Bell's theorem. Bell's theorem just underlines the fact in a way which is harder to dispute.

Naive realism means that reality is modelled by our perception, or our classical understanding, that particles always have location in space. Quantum mechanics shows that this is not true. The properties we observe (including the property of position) are not the inherent properties of particles, but they are properties of the way in which particles interact with other matter to produce our observations (and in particular the results of measurements).

Realism simply means that there exists a material reality which is responsible for our perceptions, but it does not say that this material reality is as described by our perceptions. This distinction was made as early as Plato, in his allegory of the cave. I think what you mean by an "unrealist" theory is what I would call a realist theory, but not a naive realist theory.

Perhaps the first such model was Levcippus' notion of atoms in the void. In the original form of this theory the void has no properties. In particular it contains no notion of place, or location.

Interestingly enough, this corresponds precisely to the view (held by Feynman himself) that Feynman diagrams actually do describe underlying reality. Mathematically Feynman diagrams are graphs. A graph means a collection of vertices or nodes and a collection of edges represented as lines connecting pairs of vertices. In a graph, only the connections between lines and vertices have meaning. The position on the paper of the nodes and lines is meaningless (a familiar graph is a map of the London underground, which does not show the geographical relationship of stations). Feynman diagrams are graphs showing a consistent physical structure in which space has no properties. This is a local theory in the sense that particles only have contact when they interact (as shown in vertices).

Then I think that the type of theory you are looking for is that underlying reality is described in this diagram, and the algorithms for predicting measurement results are as given in quantum electrodynamics.

enter image description here

I have given a mathematical account of this view in Mathematical_Implications_of_Relationism

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  • $\begingroup$ Our observations are a result of the way particles interact with matter. As I said in the first comments, Any world would violate Bell's inequality because his concept of particles and what they can do is very limited. Two particles can easily be correlated and tested to produce any QM prediction if you take them seriously and allow for all the proper principles. Just add one extra principle such as oscillation and you can correlate any two real objects. $\endgroup$ – Bill Alsept Dec 10 '20 at 23:17
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Short answer: the question puts into sharp focus the fact that there are various definitions of the term "local" and "realist", which are in need of careful disambiguation.

It's not possible to give a thorough treatment, but let's give a quick review of the most common definitions.

What is a local theory?

There are two main meanings given to the term local:

  1. Local in the sense of EPR and Bell. This means "no spooky action at a distance", i.e. the probabilities for the results of a process/procedure are not dependent on procedures performed far away (at space-like separations).

  2. Local in the sense of no-signaling. This means no information can be communicated faster than light from one observer to another.

Quantum theory is non-local in the first sense and local in the second. In particular, it's noteworthy that Bohm's pilot wave theory is deterministic and local (in the 2nd sense).

What is a realist theory?

Here there is even more definitional confusion. Let's list a few noteworthy definitions:

  1. Realist in the sense of preexisting information about measurement results.

This means that the system "knows" the value of any property, such as position, momentum etc. prior to measurement. This corresponds to Einstein's notion of "elements of reality".

Another way to say it would be: measurements of properties have definite numerical outcomes, which follow deterministically from the prior state of the system.

A shorthand would be to say that the theory includes hidden variables. Copenhagen quantum mechanics is not realist, but the pilot wave theory is realist in this sense.

  1. Realist in the sense of determinism.

You might be wondering how the previous definition is different from determinism. The big difference is that determinism does not presuppose that measurements of properties have definite numerical outcomes.

For example, the many world theory is deterministic, but not realist in the first sense, because measurements do not result in definite numerical outcomes, but instead in the branching of the world, with all numerical outcomes existing simultaneously.

A further distinction is that strictly speaking the first definition does not require that the time evolution between measurements be deterministic, only that measurements be deterministic.

  1. Realist in the sense of "indistinguishable from realist".

Unlike the first definition, where the theory's formalism must already include hidden variables, according to this definition a theory is realist if its predictions can be reproduced by another theory with hidden variables. According to this definition, Copenhagen quantum mechanics would be realist because its predictions are indistinguishable from those of the pilot wave theory.

Any physical theory can be reproduced by a realist2 (deterministic) theory because any source of randomness in the theory's formalism can be reproduced by a deterministic "simulation" of randomness. Therefore any physical theory that has a notion of definite numerical measurement outcomes (so not the many worlds theory) is realist in this third sense.

  1. Philosophical realism.

To satisfy this definition, the theory needs to be interpreted as describing the actual reality, as opposed to, for example, our subjective experience or epistemological state.

This is not so much an attribute of the theory itself, but a philosophical position on what a theory is, or an interpretation of the relationship between a given theory and reality.

What is local realism?

With so many definitions of the two separate terms floating around (and I certainly haven't covered them all!), it might seem like the combined term is even more ambiguous. However, the situation is not as dire. There seems to be far less disagreement on the meaning of the combined term.

A local realist theory is typically understood to be a theory which is Bell-local (i.e. local in the first sense) and realist in the first sense (measurements are not "chancy"). A shorthand would be to say that the theory includes local hidden variables.

(One alternative definition in the literature involves, instead of “normal", deterministic local hidden variables, having local but stochastic ones. I won't spend too much time on the explanation, but it's not too hard to show that this definition reduces to the theory just being Bell-local.)

What did Bell prove?

There are two opinions floating around. Bell's own opinion was that he proved that the results of quantum mechanics cannot be reproduced by a local (in the sense of Bell/Einstein, i.e. the first sense) theory.

Others think that he proved that they cannot be reproduced by a local realist theory.

Philosopher Tim Maudlin, among others, agrees with Bell. I also agree (see my other answer and comments below it for more details).

What about the original question, about a toy example of a local theory which is not realist?

Now that we have disambiguated the definitions, it should not be too hard to find such an example once we decide which exact definition we will be using.

If we decide to use the first definitions of both terms, then the Mystery Particle would be a good example of a simple local non-realist theory. This particle would be a local object that doesn't have a definite color before we shine our flashlight on it. Once we do, it randomly "chooses" the color to become/show.

Further reading.

Great reading resources have been provided by @MaximalIdeal and @gIS in the comments below the original question.

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It' important to understand the exact question:

Can anyone give me an example of a toy theory that is local but non-realist?

It seems some people have misinterpreted it as asking for just a non-realist theory, but the question crucially asks that the example also be local.

Here is very familiar example of such a theory, which most people seem to hold to: libertarian free will. This theory says:

  1. Your decision (for example to choose between vanilla and chocolate ice cream) is caused by YOU (your brain/soul), and not by something or somebody far away - locality.

  2. Your decision is not completely deterministic. Your exact prior state doesn't completely rob you of a "free choice", there is still a chance that you will choose either option - non-realism.

The trouble with this, and any other local non-realist theory, is that it cannot be distinguished from a local theory with hidden variables. For example, maybe there is some kind of invisible deterministic computer above your head that is responsible for your seemingly purely "free", random choices.

Of course, we could come up with many other examples of local non-realistic theories, and they don’t have to involve complicated entities like free agents.

For example: suppose there is a Mystery Particle, and whenever you shine your flashlight on it you can see its color. The colors of ordinary objects are determined before you look at them, but suppose this Mystery Particle is different. Suppose that before you shine your flashlight on it it's in some specific definite state O, but the moment you shine the flashlight it probabilistically "decides" to become/show a random color.

But again, if you encountered such a local non-realist particle, you could never rule out that maybe there is some kind of hidden variable, which is responsible for the seemingly random colors.

Finally, you asked about the possibility of predicting the results of measurements:

how is a prediction made? is there an algorithm at all?

Sure, for example by observing many Mystery Particles we can learn that if we prepare it in state O, then whenever we shine a flashlight on it it will become green with probability 12.5%, red with probability 73%, etc.

ADDENDUM. This is less relevant to the actual question, but if you know about Bell’s inequalities, you might be wondering if they contradict this claim I made:

“The trouble with this, and any other local non-realist theory, is that it cannot be distinguished from a local theory with hidden variables“.

To contradict this claim, we would need to have an example of a theory with two properties:

  1. It’s local non-realist
  2. It cannot be a result of some deeper, local hidden variable theory.

So didn't Bell prove that quantum mechanics is such an example?

No, he proved that quantum mechanics has the second property, not both properties! In fact, not only did he not prove property 1, he actively advocated for its exact opposite - namely for Bohmian pilot wave theory, which is non-local and deterministic!

Note that a pair of entangled particles separated by a large distance is not described in QM by a combination of two local states, i.e. the entangled state is not a tensor product of two one-particle states. (Also note that this statement is not just an idiosyncrasy specific to the Copenhagen interpretation.)

Here is what Wikipedia has to say on the matter:

Quantum nonlocality has been experimentally verified under different physical assumptions.[1][2][3][4][5] Any physical theory that aims at superseding or replacing quantum theory should account for such experiments and therefore must also be nonlocal in this sense; quantum nonlocality is a property of the universe that is independent of our description of nature. Quantum nonlocality does not allow for faster-than-light communication,[6]

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  • $\begingroup$ "The trouble with this, and any other local non-realistic theory, is that it cannot be distinguished from a local theory with hidden variables." -- Bell inequalities? $\endgroup$ – Norbert Schuch Dec 6 '20 at 20:49
  • $\begingroup$ @NorbertSchuch No, Bell inequalities prove that the results of QM can't be reproduced by a local realist theory, which doesn't contradict the claim you quoted. Note that a pair of entangled particles separated by a large distance is not described in QM by a combination of two local states, i.e. the entangled state is not a tensor product of two one-particle states. $\endgroup$ – ReasonMeThis Dec 6 '20 at 21:02
  • $\begingroup$ Your example of a mystery particle looks like a markov process. Do you mean markov processes are non realist? $\endgroup$ – bruce smitherson Dec 7 '20 at 17:22
  • $\begingroup$ Also, your claim that a nonrealist local is not distinguishable from hidden local may or may be not true, I dont think any one has a prove of that. Actually, part om my question is that, what is it a nonrealist theory in the first place, and then figure out the answer to your thesis $\endgroup$ – bruce smitherson Dec 7 '20 at 17:29
  • $\begingroup$ @Reason Bell inequalities allow to disambiguate theories which do allow for a local hidden variable description from those which don't. Your statement which I quote claims the opposite: That it cannot be distinguished. This is incorrect. $\endgroup$ – Norbert Schuch Dec 7 '20 at 22:00

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