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As far as I know, there are three possible ways to explain violation of Bell's inequality: violation of realism, violation of locality and violation of freedom. The first two are pretty straightforward even if weird, and well discussed. Violation of freedom is mentioned less often, usually in context of superdeterminism.

But as far as I understand it, violation of "freedom" only means that the results of experiments are not independent from the will of the scientist doing the experiment. What I don't get is that why is it follows that the scientist must have no free will then, assuming we want to keep both realism and locality. If the results of experiments aren't independent from the will, that only means they depend on each other somehow. Can't it be explained not by the absence of free will, but the other way around: that the will of the scientist in some really weird occult way affects the result of experiment? I realize that it sounds like a total nonsense, but then again so does every other explanation of the Bell's inequality violation. What I'm trying to understand here is whether superdeterminism is really the only possible (not the only reasonable) choice left when we exclude violations of realism and locality.

So the question is, if we assume that the mind of the scientist measuring the spin of an electron somehow affects the result of the measurement, would that explain Bell's inequality violation without the need for either nonlocality, nonrealism or superdeterminism?

Or, in even more general case, without introducing any kind of occult mystics: to explain Bell's inequality violation while still keeping some sort of hidden local variable theory, is it not enough to introduce some kind of dependence between the choice of a particular axis to measure the spin along and the result of the measurement? It is called hidden variable after all, so it could be changing depending on something.

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  • $\begingroup$ Perhaps the "will" of the experimenters is manifested in their reluctance to notice that there is nothing weird about the fact that entangled particles remain entangled. $\endgroup$ – bright magus Oct 18 '14 at 11:44
  • $\begingroup$ @bright magus, nothing weird in common sense (Bell's inequality follows from it) being so openly violated? Entanglement itself is quite weird, whether it remains or not. $\endgroup$ – Sergei Tachenov Oct 19 '14 at 13:00
  • $\begingroup$ Common sense would rather expect the entangled particles to remain entangled until something causes them not to be. The particles simply retain the relation between their spins after they separate, that's all there is to it. $\endgroup$ – bright magus Oct 19 '14 at 13:14
  • $\begingroup$ @bright, common sense doesn't exactly work that way. If we assume that this relation actually exists, then spins themselves must exist (be defined) as well. But then Bell's inequality must be true. If we assume that spins are random, then common sense says that the relation itself can't possibly exist. $\endgroup$ – Sergei Tachenov Oct 19 '14 at 18:42
  • $\begingroup$ "But then Bell's inequality must be true." Let's talk about bias. Why does the equation have three plus terms and one minus? Why not two and two? Why asking? There is no such thing as "simultaneous detection", because it assumes infinite precision which doesn't exist. So there must be some margin of error set up in the machine. But then, if you add 3 terms and subtract one, the sum will always be above 2. Always. Also, QM predicts 2.828, right? The actual numbers obtained are rarely revealed, but I urge you to look them up and see what they are closer to. Especially the early ones ... $\endgroup$ – bright magus Oct 19 '14 at 19:14
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I don't think this really makes sense. Here's an analogy I wrote up once to illustrate one of the simplest Bell inequalities:

Suppose we have a machine that generates pairs of scratch lotto cards, each of which has three boxes that, when scratched, can reveal either a cherry or a lemon. We give one card to Alice and one to Bob, and each scratches only one of the three boxes. When we repeat this many times, we find that whenever they both pick the same box to scratch, they always get the same result--if Bob scratches box A and finds a cherry, and Alice scratches box A on her card, she's guaranteed to find a cherry too.

Classically, we might explain this by supposing that there is definitely either a cherry or a lemon in each box, even though we don't reveal it until we scratch it, and that the machine prints pairs of cards in such a way that the "hidden" fruit in a given box of one card always matches the hidden fruit in the same box of the other card. If we represent cherries as + and lemons as -, so that a B+ card would represent one where box B's hidden fruit is a cherry, then the classical assumption is that each card's +'s and -'s are the same as the other--if the first card was created with hidden fruits A+,B+,C-, then the other card must also have been created with the hidden fruits A+,B+,C-.

The problem is that if this were true, it would force you to the conclusion that if Alice and Bob are picking randomly which box to scratch on each trial (with a 1/3 chance of A, B, or C each time), then if they do this a large number of times, we should expect that in the subset of trials where Alice and Bob happened to pick different boxes to scratch, they should find the same fruit at least 1/3 of the time. For example, if we imagine Bob and Alice's cards each have the hidden fruits A+,B-,C+, then we can look at each possible way that Alice and Bob can randomly choose different boxes to scratch, and what the results would be:

Bob picks A, Alice picks B: opposite results (Bob gets a cherry, Alice gets a lemon)

Bob picks A, Alice picks C: same results (Bob gets a cherry, Alice gets a cherry)

Bob picks B, Alice picks A: opposite (Bob gets a lemon, Alice gets a cherry)

Bob picks B, Alice picks C: opposite results (Bob gets a lemon, Alice gets a cherry)

Bob picks C, Alice picks A: same results (Bob gets a cherry, Alice gets a cherry)

Bob picks C, Alice picks picks B: opposite results (Bob gets a cherry, Alice gets a lemon)

In this case, you can see that that if they are equally likely to pick each combination of boxes, then 2 times out of 6 when they choose different boxes, they will get the same fruit (i.e. a 1/3 chance of the same result). You'd get the same answer if you assumed any other preexisting state where there are two fruits of one type and one of the other, like A+,B+,C- or A+,B-,C-. On the other hand, if you assume a state where each card has the same fruit behind all three boxes, so either they're both getting A+,B+,C+ or they're both getting A-,B-,C-, then of course even if Alice and Bob pick different boxes to scratch they're guaranteed to get the same fruits with probability 1. So if you imagine that when multiple pairs of cards are generated by the machine, some fraction of pairs are created in inhomogoneous preexisting states like A+,B-,C- while other pairs are created in homogoneous preexisting states like A+,B+,C+, then the probability of getting the same fruits when you scratch different boxes should be somewhere between 1/3 and 1. 1/3 is the lower bound, though--even if 100% of all the pairs were created in inhomogoneous preexisting states, it wouldn't make sense for you to get the same answers in less than 1/3 of trials where you scratch different boxes, provided you assume that each card has such a preexisting state with "hidden fruits" in each box.

But now suppose Alice and Bob look at all the trials where they picked different boxes, and found that they only got the same fruits 1/4 of the time! That would be the violation of Bell's inequality, and something equivalent actually can happen when you measure the spin of entangled photons along one of three different possible axes. So in this example, it seems we can't resolve the mystery by just assuming the machine creates two cards with definite "hidden fruits" behind each box, such that the two cards always have the same fruits in a given box.

Something mathematically analogous is predicted in certain experiments where entangled photons are passed through polarizers at different angles, in which the probability that both photons will have the same result (both pass through their respective polarizers, or both are blocked) is $ cos^2 (\theta) $, where $ \theta $ is the angle between the two polarizers. Suppose the experimenters decide in advance that on each trial, they will choose randomly between one of three angles for their polarizer: 0 degrees from vertical, 60 degrees, or 120 degrees. On a given trial, if both experimenters choose the same angle, then $ \theta = 0 $ so they are guaranteed to get the same result with probability 1 (both photons pass or both are blocked), but if they choose different settings the probability of getting the same result would be $ cos^2 (\pm 60) $ or $ cos^2 (\pm 120) $, which in both cases gives a probability of 1/4. But the Bell inequality whose derivation I sketched says that if you want to explain the perfect match when both experimenters make the same choice using local hidden variables, the probability of a match when they make different choices should be no lower than 1/3.

In terms of this analogy, here's how "violation of locality" or "violation of freedom" could explain the results that when both experimenters make the same measurement they get the same result with probability 1, but when they make different measurements they only get the same result 1/4 of the time. (I won't attempt to make an analogy for "violation of realism" because I don't really understand what that means, although some authors like this one seem to treat it as synonymous with violation of the experimenter's choice being statistically independent of hidden variables, which is just 'violation of freedom')

Violation of locality: Suppose that instead of a lotto card with preprinted hidden fruit under each square, the two experimenters instead receive touchscreen devices with three gray squares A, B, and C, and they can choose which square to tap, transforming that square into a picture of a fruit. If there is an FTL communication link between the two devices, and there is a brief interval between each experimenter tapping a square and the device displaying a fruit, then the computing elements inside the two devices can pool the information about what square each experimenter should see--if both experimenters tap the same square the devices can be programmed to show the same fruit, and if they tap different squares, the devices can be programmed to decide using a random or pseudorandom algorithm that gives a 1/4 chance of both displaying the same fruit and a 3/4 chance of displaying opposite fruits.

Violation of freedom: We could imagine that at the time the machine prints the two cards (or earlier), some entity sends a subliminal signal to each experimenter which predetermines what choice they will make about what square to scratch later, and also determines what fruits are printed under each square in a way that's correlated with the experimenter's choices. So on each trial where this entity predetermines the two experimenters to scratch the same square, it can also cause the card-printing machine to create a pair of cards with the same fruit under that square on each card; and on the trials where the entity predetermines that the experimenters scratch different squares, on 1/4 of those trials it can cause the machine to create pairs with the same fruit under the two squares that will be scratched, and on 3/4 it can cause the machine to create pairs with different fruits.

A variant of this would be a deterministic universe where the initial conditions are chosen very carefully to ensure the above sort of correlation between the hidden fruits sent out on each trial and which boxes the experimenters choose to scratch on each trial. The assumption that there is no "conspiracy" in the initial conditions of the universe which predetermines a strong correlation between the hidden variables associated with particles on each trial (or hidden fruits on each pair of cards in my analogy) and what variables the experimenters will choose to measure on each trial (or what boxes they will choose to scratch) is often called the "no conspiracy" assumption, see the paper on the detailed assumptions of Bell's theorem here which discusses it in section D on page 6. Models with violations of no-conspiracy that depend on fine-tuning of initial conditions are sometimes grouped under the label "superdeterminism", see here and here for example.

Another subtle variation on this is the idea that the entity doesn't actually cause the experimenters to make particular choices but merely has a sort of "precognition" about what choices they will in fact make (implying some backwards-in-time causality), and determines what fruits will be under the two boxes that it knows will be scratched in the future using the same type of rule as above. The philosopher of science Huw Price discusses such an idea in his book Time's Arrow and Archimedes' Point.

(As I mentioned in a comment above, advocates of the many-world interpretation also make the point that one can preserve locality if one drops the assumption that each measurement gives a single unique result, as elaborated in this paper by David Deutsch. If you want a simple conceptual toy model of how this could work, see this post I wrote up on physicsforums.com a while ago).

Thinking in these terms, your idea about the experimenter's "will" affecting the experiment could be imagined in terms of the scenario where each experimenter receives a touchscreen device which "decides" what fruit to reveal on-the-fly after the experimenter has tapped one of the three squares. We could imagine that the device has a built-in psychic receiver that's sensitive to the experimenter's will--consciously or unconsciously, the experimenter decides what fruit she wants to see when she hits a given square, and the device obliges by showing exactly that fruit. To make it more concrete we could also just drop the notion of psychic powers, and just imagine that each experimenter receives a touchscreen device in which they first tap a square, then there's a pop-up menu allowing them to decide what fruit will appear in that square's place. Without doing a detailed mathematical derivation in the style of Bell's, it's easy enough to play around with this and see that as long as each experimenter is choosing the square randomly and has no knowledge of what square the other experimenter chose, there is no classical way for them to both guarantee that they both get the same fruit whenever they tap the same square, and to guarantee that they only get the same fruit 1/4 of the time whenever they tap different squares. They could only do the first if they had agreed in advance something like "regardless of which square we choose, let's both make the result be a lemon on this trial", but that would mean that even if they pick different squares they would both get the same fruit. On the other hand, if they had made no agreement in advance about what fruit to "will" and couldn't communicate once they were moved to separate locations, then their ability to choose the fruit wouldn't help them to guarantee that they reveal the same fruit when they choose the same square.

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    $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – David Z Oct 14 '14 at 20:02
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The experimenter is part of the measuring apparatus. You can't circumvent Bell's inequality by changing the way the apparatus works.

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