What do violations of Bell's inequalities tell us about nature?

[ttn]

... who needs additional magic like non-locality at all costs and what does explain better?

What non-locality explains better is the results of the Aspect (-type) experiments. The data from these experiments cannot be explained by any local theory. That is what Bell proved.

Don't agree? Then please please please address the challenge I keep posting: tell me how to explain even just the one simple subset of the data (namely, that there are perfect correlations when Alice and Bob measure along the same axis)

(You I think mean to be pointing out that we already know that "realism" is false. Presumably you are thinking of the Kochen-Specker and other similar "no-hidden-variable" theorems. I agree. Realism in that sense is already known to be false. But as bohm2 has explained, this is just a red herring here. To say that Bell's theorem does not prove nonlocality because we already know that realism is false, is like saying that the Earth doesn't go around the sun because we already know the Earth is round. It is just a total non-sequitur. It is possible to know more than one thing, so discovering X does not in any way preclude or automatically refute a purported later proof of Y.)

Though it seems obvious that if realism fails, so does locality and nonlocality is implied by the consistence of the classical world and in the end both will be found to be incorrect and incompatible with qm.

So you agree that no local theory is consistent with experimental data? I can't exactly follow the comments here.

 

[ttn]

ttn: here is the paper by Wallace where he describes some of the ideas regarding ontology: http://arxiv.org/abs/1111.2189

Thanks. I actually just ordered his book and am looking forward to reading it, but I'll check this out too.

 

[harrylin]

[..] Tell me how what you're saying isn't just parallel to that (I think, manifestly absurd) response to the hypothetical scenario.

Not sure what he means, but it doesn't seem fair to give him/her the disadvantage of the doubt; fair would be to give him/her the advantage of the doubt. For example, it could be more similar to MMX type experiments: contrary to the generalizing nonsense that one sometimes reads about it those merely disproved a specific set of hypotheses that was put to the test.

[..] QM is a nonlocal theory, at least by the best definition of locality that we have going -- namely, Bell's as presented in "la nouvelle cuisine". You have a better/different formulation of "locality" to propose? I'm all ears. Or you think there's some flaw in Bell's formulation? I'm all ears. [..]

See the recent discussion here:
https://www.physicsforums.com/showthread.php?p=4236787
Thus, apparently Bell implied with "realism" that it "is meaningful to assign a property to a system (e.g. the position of an electron) independently of whether the measurement of such property is carried out."
IMHO that fundamentally disagrees with QM while it is not required for the concept of physical reality, as an electron could be extended as a wave without having a precise, single position. That it should have such a position is an unrealistic definition of "realism" as it only corresponds to a specific subset of models of reality.

 

[ttn]

See the recent discussion here:
https://www.physicsforums.com/showthread.php?p=4236787
Thus, apparently Bell implied with "realism" that it "is meaningful to assign a property to a system (e.g. the position of an electron) independently of whether the measurement of such property is carried out."
IMHO that fundamentally disagrees with QM while it is not required for the concept of physical reality, as an electron could be extended as a wave without having a precise, single position. That it should have such a position is an unrealistic definition of "realism" as it only corresponds to a specific subset of models of reality.

What part of that thread is supposed to be relevant? I mean, obviously, there are similar issues being discussed, but I don't know what specifically you meant to be pointing to.

I don't understand your sentence starting "Thus, apparently Bell implied..." Who are you quoting there? I'm pretty sure that isn't a statement of Bell's! Indeed, I don't recall Bell ever talking about "realism". The whole idea that "realism" is somehow relevant to Bell's theorem is an invention of the people who haven't actually studied/understood Bell.

I think I agree with your last couple of sentences, but again isn't the point just that "realism" is used to mean a number of rather different things, so people should be careful to define exactly what they mean whenever they use the term? For example, if the goal is to have what is sometimes described as a "realist interpretation of QM" -- that is, some kind of good old-fashioned style physics theory that makes postulates about what kind of stuff exists and how it acts (rather than, e.g., operationalist-style postulates about laboratory procedures) -- then, no doubt, it would be ridiculous to demand from the outset that electrons must have definite sharp positions at all times. Maybe it will turn out that electrons are like little fuzzy clouds, or like groups of ripples on a pond. Such models would be perfectly "realist" in this sense and certainly shouldn't be ruled out a priori at the outset. I think that was your point, and I totally agree.

But I think people who voted for "anti-realism" in the poll did *not* mean that *this* sort of "realism" is refuted by Bell's theorem. They meant instead the idea that there should exist deterministic non-contextual hidden variables for all (?) "observables" recognized by QM. They are correct that this other sort of "realism" is indeed false, but they are wrong to think that this is the lesson of Bell's theorem. We already knew this realism was wrong, from von Neumann, Kochen-Specker, etc. Bell taught something new, something that has nothing to do with realism.

Anybody who disagrees should explain how to account for the perfect correlations in a local but non-realist way. (Everybody knows you can explain the perfect correlations in a local way *with* this deterministic non-contextual HV sense of "realism". But then everybody also knows that you can't explain the *rest* of the QM predictions with that kind of theory. The question is whether the QM predictions can be accounted for by a theory that is local but non-realist. The challenge is to show that such a model can even account for the perfect correlations subset of the QM predictions. No takers so far, unfortunately.)

 

[rubi]

QM is a nonlocal theory, at least by the best definition of locality that we have going -- namely, Bell's as presented in "la nouvelle cuisine". You have a better/different formulation of "locality" to propose? I'm all ears. Or you think there's some flaw in Bell's formulation? I'm all ears.

Bell's definition of "locality" just can't be applied to every possible theory of nature, because it assumes that nature can always be described by classical probabilities of the form p(a,b,\lambda), where \lambda are some unknown parameters. This requirement isn't fulfilled in quantum mechanics, so this definition of locality can't be applied to it in the first place. Quantum mechanics neglects the existence of functions p(a,b,\lambda) entirely. It only predicts functions of the type p(a,b). So it isn't even possible to decide whether QM is Bell-local or not. In other words: Bells definition of locality isn't general enough to cover all possible theories of nature (including QM) and thus isn't a useful criterion to classify theories at all.

A useful criterion to classify "locality" that covers both QM and classical probability theories is this: A theory is local if an event in one region of spacetime can't influence the experimental outcomes of an experiment in a spacelike separeted region.

In that sense, QM predictions can be explained by completely local quantum theories. Of course non-relativistic QM doesn't count, but relativistic theories like Wightman QFT's can explain the predictions. Locality is even an axiom there.

If you worry about the nonlocal correlations of QM, let's make a simple gedankenexperiment:

You have a green ball and a red ball and put them in two identical boxes. You send these boxes to two different people. These people know that you started with a green and a red ball. So the probability to get green/red is 1/2. When person 1 opens his box, he will get a definite result. Let's say he gets red. Then he knows immediately that person 2 has the green ball in his box, even if that box hasn't been opened yet. This is definitely a nonlocal correlation, but nobody would consider this as an action at a distance.

Up to now, this isn't quantum mechanics yet. But let's do the same experiment with qubits instead of bits. Instead of green and red balls, we put particles with spin into these boxes. We create 2 particled with orthogonal spin states, put them in the boxes and repeat the same experiment. Of course we get nonlocal correlations again, because we separated two particles that were created with correlation locally.

So all the weirdness concerning "nonlocality" is gone and what remains is the standard QM weirdness about the existence of superpositions of states.

 

[harrylin]

What part of that thread is supposed to be relevant? I mean, obviously, there are similar issues being discussed, but I don't know what specifically you meant to be pointing to.

I don't understand your sentence starting "Thus, apparently Bell implied..." Who are you quoting there? [..]

That post (not that thread) refers to the clarification by DrChinese in post #49 there that Bell imposes the requirement to realism of post #48 that was brought up in the discussion and which for your convenience I also cited here. As discussed there, Bell didn't make that sufficiently clear.

Indeed, I don't recall Bell ever talking about "realism". The whole idea that "realism" is somehow relevant to Bell's theorem is an invention of the people who haven't actually studied/understood Bell.

In view of those remarks of yours, DrChinese's clarification there as well as Bell's paper "Bertlmann's socks and the nature of reality" will be interesting for you.

if the goal is to have what is sometimes described as a "realist interpretation of QM" -- that is, some kind of good old-fashioned style physics theory that makes postulates about what kind of stuff exists and how it acts (rather than, e.g., operationalist-style postulates about laboratory procedures) -- then, no doubt, it would be ridiculous to demand from the outset that electrons must have definite sharp positions at all times. Maybe it will turn out that electrons are like little fuzzy clouds, or like groups of ripples on a pond. Such models would be perfectly "realist" in this sense and certainly shouldn't be ruled out a priori at the outset. I think that was your point, and I totally agree.

But I think people who voted for "anti-realism" in the poll did *not* mean that *this* sort of "realism" is refuted by Bell's theorem. They meant instead the idea that there should exist deterministic non-contextual hidden variables for all (?) "observables" recognized by QM. They are correct that this other sort of "realism" is indeed false, but they are wrong to think that this is the lesson of Bell's theorem. We already knew this realism was wrong, from von Neumann, Kochen-Specker, etc. Bell taught something new, something that has nothing to do with realism.

The issue that recently came up in the linked thread is that those two things are perhaps not unrelated; and in the abovementioned discourse Bell himself highlighted that his theorem has much to do with a certain "realism". Even, as it turns out, "counterfactual" realism. I don't know if it makes a difference but for the moment I'm not convinced about anything.

Anybody who disagrees should explain how to account for the perfect correlations in a local but non-realist way. [..]

Suppose I don't know how a TV works but that I can believe that TV could work. Now you say that therefore I should believe that TV cannot work? That doesn't make any sense to me. A puzzle is just a puzzle, not a conclusion.

The question is whether the QM predictions can be accounted for by a theory that is local but non-realist. The challenge is to show that such a model can even account for the perfect correlations subset of the QM predictions. No takers so far, unfortunately.)

Apparently Neumaier is a taker for a realistic explanation that you call "non-realist" (post #53 there), but not yet ready to deliver. I'm an interested onlooker.

 

[harrylin]

Bell's definition of "locality" just can't be applied to every possible theory of nature, because it assumes that nature can always be described by classical probabilities of the form p(a,b,\lambda), where \lambda are some unknown parameters. This requirement isn't fulfilled in quantum mechanics, so this definition of locality can't be applied to it in the first place. Quantum mechanics neglects the existence of functions p(a,b,\lambda) entirely. It only predicts functions of the type p(a,b). [..] we put particles with spin into these boxes. We create 2 particled with orthogonal spin states, put them in the boxes and repeat the same experiment. Of course we get nonlocal correlations again, because we separated two particles that were created with correlation locally.

So all the weirdness concerning "nonlocality" is gone and what remains is the standard QM weirdness about the existence of superpositions of states.

We have had earlier discussions about that, related to Jaynes -it sounds as if you are referring to him. However that's a bit too simplistic, as I found out myself when I presented his arguments here (you can search this forum for Jaynes). There could be something to it, perhaps related to unknown possible type of models, but that never came out as far as I am aware of. While technically speaking his argument is correct (IMHO), it doesn't seem to cut wood. If there is something substantial to it, it still has to be presented on this forum.

 

[stevendaryl]

Bell's definition of "locality" just can't be applied to every possible theory of nature, because it assumes that nature can always be described by classical probabilities of the form p(a,b,\lambda), where \lambda are some unknown parameters. This requirement isn't fulfilled in quantum mechanics, so this definition of locality can't be applied to it in the first place.

I think you're just repeating what Bell's theorem proves, which is that the quantum predictions for probabilities can't be reproduced by a "local variables" theory.

 

[ttn]

Bell's definition of "locality" just can't be applied to every possible theory of nature, because it assumes that nature can always be described by classical probabilities of the form p(a,b,\lambda), where \lambda are some unknown parameters. This requirement isn't fulfilled in quantum mechanics, so this definition of locality can't be applied to it in the first place. Quantum mechanics neglects the existence of functions p(a,b,\lambda) entirely. It only predicts functions of the type p(a,b). So it isn't even possible to decide whether QM is Bell-local or not. In other words: Bells definition of locality isn't general enough to cover all possible theories of nature (including QM) and thus isn't a useful criterion to classify theories at all.

You're just factually wrong here. Ordinary QM absolutely does assert probabilities of the form p(a,b,\lambda) -- it's just that "λ", for QM, is nothing but the wave function ψ. You are of course thinking "no, no, λ is supposed to represent a hidden variable, and by definition there are no such things in QM". But you're just mistaken about how Bell's formulation of locality works. There is nothing like an assumption that "λ" must represent some part of a state description that *supplements* the quantum wave function. The "λ" is rather simply meant to denote "whatever some candidate theory says a complete specification of the physical state of the particle pair, prior to measurement, consists of". So, for ordinary QM, λ is just ψ. For the pilot wave theory, λ is the wave function + the particle positions. And so on.

Go read Bell (start with "la nouvelle cuisine") if you don't believe me.

A useful criterion to classify "locality" that covers both QM and classical probability theories is this: A theory is local if an event in one region of spacetime can't influence the experimental outcomes of an experiment in a spacelike separeted region.

I agree, that's a nice formulation. But how exactly do you translate it into a sharp mathematical statement? How exactly do we decide what it means for one event to "influence" another? What exactly do/should we mean by influencing the "outcomes" -- does this mean only the *statistics* of outcomes, or does it mean the outcomes (or their probabilities) for an actual individual case, or what? The point is: Bell has already worried about these issues and answered these questions! His formulation of locality (in "la nouvelle cuisine") is precisely the needed thing -- a sharp mathematical statement of what you formulate here in words.

In that sense, QM predictions can be explained by completely local quantum theories.

Not true. Even in the simple case of Alice and Bob measuring spin/polarization along parallel axes, a=b, QM's account of the empirical correlations is nonlocal: (supposing Alice happens to measure hers first, then) Alice's measurement influences the state of Bob's particle, which in turn affects Bob's outcomes. (Certain outcomes that were possible -- P = 50% -- prior to Alice's measurement, now become impossible -- P=0 -- for example.)

Of course non-relativistic QM doesn't count, but relativistic theories like Wightman QFT's can explain the predictions. Locality is even an axiom there.

The "locality" that is sometimes taken as an axiom in QFT is different from the "locality" that is at issue in Bell's theorem. The former actually just amounts to what is usually called "no signalling" in the Bell literature. But we know (from the concrete example of the dBB pilot-wave theory for example) that theories can be blatantly non-local (in the Bell sense) and yet be perfectly "local" in the no-signalling sense (because the hidden variables aren't accessible or controllable or whatever).

You have a green ball and a red ball and put them in two identical boxes. You send these boxes to two different people. These people know that you started with a green and a red ball. So the probability to get green/red is 1/2. When person 1 opens his box, he will get a definite result. Let's say he gets red. Then he knows immediately that person 2 has the green ball in his box, even if that box hasn't been opened yet. This is definitely a nonlocal correlation, but nobody would consider this as an action at a distance.

I agree, there's no nonlocality there. Incidentally, a good homework problem would be: go study Bell's formulation of "locality" until you can explain precisely how to use Bell's formulation to (correctly!) diagnose this situation as not involving any nonlocality. This is exactly the kind of exercise one must go through to convince oneself that Bell's formulation is a good formulation!

Up to now, this isn't quantum mechanics yet. But let's do the same experiment with qubits instead of bits. Instead of green and red balls, we put particles with spin into these boxes. We create 2 particled with orthogonal spin states, put them in the boxes and repeat the same experiment. Of course we get nonlocal correlations again, because we separated two particles that were created with correlation locally.

So all the weirdness concerning "nonlocality" is gone and what remains is the standard QM weirdness about the existence of superpositions of states.

I certainly agree that this is exactly the right concrete example to be thinking about to make all these issues crystal clear! But I think you aren't yet there, because you haven't yet understood/appreciated Bell's definition of locality. So, seriously, go read Bell's paper. Then you'll see exactly how, actually, in this situation (I assume here you have in mind that the two particles should be in the total spin zero, the "singlet", state) one can see unambiguously that ordinary QM is nonlocal. It comes down to this: even conditionalizing on what ordinary QM says the complete state of the particles prior to measurement is, the probability P that the theory attributes to a certain one of the possible outcomes (say, B) for Bob's measurement *depends* on the outcome of Alice's measurement (A): P(B|a,b,A,λ) is not equal to P(B|a,b,λ) ... even though the event "A" is spacelike separated from "B". So, translating back to ordinary language, we'd say that A is influencing the outcome B (or more precisely, the probability distribution over the possible outcomes, since we are specifically avoiding any assumption of determinism).

Crucial note: what this proves is that *ordinary QM's explanation of the correlations is nonlocal*. This is not the same as saying, for example, that the correlations that occur when a=b prove the real existence (in nature, not just some candidate theory) of nonlocality. Indeed, we know that the perfect correlations for the case a=b *can* be explained locally -- but *supplementing* QM's λ (namely, ψ) with some additional "hidden variables" that pre-determine the outcome. That was pointed out long ago by Einstein. Bell's discovery was that such models cannot account for the more general correlations that occur when a =/= b.

 

[ttn]

That post (not that thread) refers to the clarification by DrChinese in post #49 there that Bell imposes the requirement to realism of post #48 that was brought up in the discussion and which for your convenience I also cited here. As discussed there, Bell didn't make that sufficiently clear.

Thanks for the clarification of what you were pointing to. I am familiar with Dr Chinese' unique interpretation of these sentences from Bell. But actually it is not Bell here who failed to make something clear -- it is rather Dr C who totally misunderstands the issue. There is absolutely no *assumption* of (what Dr C means by) "realism" in Bell's 1964 paper. And Bell makes this even clearer in his many subsequent papers. The relevant money quote here is something I partially quoted earlier in this thread, from the B's sox paper: "It is remarkably difficult to get this point across, that determinism [aka, DrC's "realism"] is not a *presupposition* of the analysis. There is a widespread and erroneous conviction that for Einstein {^10} determinism was always *the* sacred principle. The quotability of his famous 'God does not play dice' has not helped in this respect... [Bell then here gives some quotes from Born which exhibit this "erroneous conviction". He then continues:] Misunderstanding could hardly be more complete. Einstein had no difficulty accepting that affairs in different places could be correlated. What he could not accept was that an intervention at one place could *influence*, immediately, affairs at another."

And pay special attention to the footnote {^10}: "And his followers. [i.e., Bell is describing himself, as one of Einstein's followers.] My own first paper on this subject (Physics 1, 195 (1965)) starts with a summary of the EPR argument *from locality to* deterministic hidden variables. But the commentators have almost universally reported that it begins with deterministic hidden variables."

And that is precisely the error made by everybody who voted for "anti-realism" in the poll here. They simply *miss* that the argument begins with the EPR argument *from locality to* "realism". They look only at the *second* part of the argument, which shows that "realism" + locality implies a contradiction with experiment. So they *mistakenly* think that we get to choose which of "realism" or locality to reject, in order to avoid the conflict. But in fact there is no such choice. Locality already entails "realism". So to have to choose one to reject is to have to choose locality.

In view of those remarks of yours, DrChinese's clarification there as well as Bell's paper "Bertlmann's socks and the nature of reality" will be interesting for you.

I think the sock is actually on the other foot.

The issue that recently came up in the linked thread is that those two things are perhaps not unrelated; and in the abovementioned discourse Bell himself highlighted that his theorem has much to do with a certain "realism". Even, as it turns out, "counterfactual" realism. I don't know if it makes a difference but for the moment I'm not convinced about anything.

Fair enough. Hopefully the passage I quoted above will encourage everybody to go back and revisit this indeed excellent paper of Bell's. Here is another good passage from it which stresses *exactly* the relevant point here:

"Let us summarize once again the logic that leads to the impasse [i.e., the conclusion of nonlocality]. The EPRB correlations are such that the result of the experiment on one side immediately foretells that on the other, whenever the analyzers happen to be parallel. If we do not accept the intervention on one side as a causal influence on the other, we seem obliged to admit that the results on both sides are determined in advance anyway, independently of the intervention on the other side, by signals from the source and by the local magnet setting. [That was his recapitulation of the EPR argument. Now he turns to the second part of the 2-part argument, the part that is often misleadingly called "Bell's theorem".] But this has implications for non-parallel settings which conflict with those of quantum mechanics. So we *cannot* dismiss intervention on one side as a causal influence on the other."

Apparently Neumaier is a taker for a realistic explanation that you call "non-realist" (post #53 there), but not yet ready to deliver. I'm an interested onlooker.

Me too!

 

[stevendaryl]

Not true. Even in the simple case of Alice and Bob measuring spin/polarization along parallel axes, a=b, QM's account of the empirical correlations is nonlocal: (supposing Alice happens to measure hers first, then) Alice's measurement influences the state of Bob's particle, which in turn affects Bob's outcomes. (Certain outcomes that were possible -- P = 50% -- prior to Alice's measurement, now become impossible -- P=0 -- for example.)

I don't think that you can prove that Alice's measurement influences Bob. In ordinary probability theory, someone puts a ball into either Bob's box, or Alice's box, but you don't know which. When Alice opens her box, the probability of Bob finding the ball either drops from 50% to 0%, or rises from 50% to 100%. Does that mean that Alice influenced Bob's box? No, because probabilities are not treated realistically--they are interpreted as reflecting lack of information, or something, rather than something objective about the world.

If you interpret probabilities realistically, then quantum mechanics is nonlocal, but so is classical probability theory. So that's too loose a notion of "nolocal".

The "locality" that is sometimes taken as an axiom in QFT is different from the "locality" that is at issue in Bell's theorem. The former actually just amounts to what is usually called "no signalling" in the Bell literature. But we know (from the concrete example of the dBB pilot-wave theory for example) that theories can be blatantly non-local (in the Bell sense) and yet be perfectly "local" in the no-signalling sense (because the hidden variables aren't accessible or controllable or whatever).

I think that the "no signalling" criterion is the most objective way to define locality. In a Bohm-type theory, there ARE nonlocal interactions, which through a conspiracy manage to be undetectable and unusable for FTL signalling. That's sort of like the case with certain sophisticated aether models for electromagnetic interactions. In these models, there is an absolute rest frame, but things conspire to make it impossible to detect this frame.

I think that both an aether model and a Bohm model are aesthetically unappealing, in exactly the same way:they introduce things into the ontology (absolute rest frame, faster-than-light interactions), and then introduce laws of physics that make these things unobservable. That seems like an unnecessary complication---if there is an element of the model that is unobservable, then it seems better to leave it out, if at all possible.

 

[rubi]

You're just factually wrong here. Ordinary QM absolutely does assert probabilities of the form p(a,b,\lambda) -- it's just that "λ", for QM, is nothing but the wave function ψ.

I agree that if there were probabilities of the form p(a,b,\lambda), then for QM, \lambda would have to be the wave function. However, the space of wave functions is too big to be a probability space in the sense of probability theory. In more rigorous terms, this means that no one has succeeded in specifying a sigma-algebra and a probabilty measure on sufficiently big spaces of wave-functions (like the spaces used in QM, for example L^2(R) or Fock spaces). Thus your p(a,b,\psi)'s can't be probabilities and thus my claim still holds: This definition of locality can't be applied to QM.

I agree, that's a nice formulation. But how exactly do you translate it into a sharp mathematical statement?

You can't make a general formal definition, because different theories have different frameworks that work differently. There is no definition that can cover all possible theories (at least i don't know one). You have to look at your given theory and make up a definition for locality that ensures that your intuitive understanding is fulfilled. In Wightman QFT for example, you postulate that commutators of spacelike separated obervables vanish.

Not true. Even in the simple case of Alice and Bob measuring spin/polarization along parallel axes, a=b, QM's account of the empirical correlations is nonlocal: (supposing Alice happens to measure hers first, then) Alice's measurement influences the state of Bob's particle, which in turn affects Bob's outcomes. (Certain outcomes that were possible -- P = 50% -- prior to Alice's measurement, now become impossible -- P=0 -- for example.)

But if that experiment is performed a hundred times, Bob gets a completely consistent probability distribution that he would also calculate if his particle wouldn't be entangled with Alice's particle.

To make the point more clear: Let's assume every particle of the Earth were entangled with some particle in the andromeda galaxy. Is there a way to find out without traveling to andromeda?

 

[rubi]

We have had earlier discussions about that, related to Jaynes -it sounds as if you are referring to him. However that's a bit too simplistic, as I found out myself when I presented his arguments here (you can search this forum for Jaynes). There could be something to it, perhaps related to unknown possible type of models, but that never came out as far as I am aware of. While technically speaking his argument is correct (IMHO), it doesn't seem to cut wood. If there is something substantial to it, it still has to be presented on this forum.

I'm not referring to Jaynes. I've searched the forums, but unfortunately i there's too much results for me to look through. Can you point me to the thread you are referring to?

Maybe i need to point out that i don't object Bell's or CHSH's (and so on) theorems. They're completely valid.

 

[ttn]

I don't think that you can prove that Alice's measurement influences Bob. In ordinary probability theory, someone puts a ball into either Bob's box, or Alice's box, but you don't know which. When Alice opens her box, the probability of Bob finding the ball either drops from 50% to 0%, or rises from 50% to 100%. Does that mean that Alice influenced Bob's box? No, because probabilities are not treated realistically--they are interpreted as reflecting lack of information, or something, rather than something objective about the world.

Well, I agree (and indeed wrote!) that (again, considering the simple case where a=b) you cannot prove that Alice's measurement influences Bob. There are local ways to explain these perfectly correlations! And there are nonlocal ways to explain them! Who knows which is right. But the point is, Bell's definition of locality gives us a way to assess whether a *particular candidate theory* (say, ordinary QM, or the pilot wave theory, or some "local realist" theory, ...) provides a local explanation or not. So your response above, while sensible, is actually a response to an assertion that wasn't made. The specific assertion was that *ordinary QM's account, for the perfect a=b correlations, is nonlocal*. Yes, this does not preclude the existence of different theories perhaps explaining those same correlations in a local way. And so it doesn't prove that nonlocality is required to explain them. But just because some *other* theory can explain those particular correlations locally, doesn't mean ordinary QM does! It doesn't!

But again, the lesson here is that if you are slightly confused about all this stuff, then it just means there is a really important and cool thing out there -- namely Bell's formulation of locality -- that you haven't appreciated yet. So go read his paper, or mine, or something.

If you interpret probabilities realistically, then quantum mechanics is nonlocal, but so is classical probability theory. So that's too loose a notion of "nolocal".

Neither Bell or I holds the notion of locality you have in mind here.

I think that the "no signalling" criterion is the most objective way to define locality. In a Bohm-type theory, there ARE nonlocal interactions, which through a conspiracy manage to be undetectable and unusable for FTL signalling.

Now that's an interesting set of statements! So, you agree that, for whatever reason, the Bohm theory precludes signalling (i.e., basically, it agrees with the empirical predictions of QM, including that Bob's marginal shouldn't be affected by Alice's setting). And you want "locality" to just *mean* this no-signalling condition. But then... what in the world do you mean when you say that, despite the no signalling, there ARE nonlocal interactions in the Bohm theory? I actually agree with you that there are, and that this is somehow so obvious that nobody who knew what they were talking about could disagree with it. But didn't you just say there is no objective meaning to "locality" other than the no-signalling business? So, seriously, what do you mean when you say that Bohm's theory is obviously nonlocal? Or is it that we should define locality one way when we're looking at Bohm's theory (you know, to make sure we come to the right conclusion, namely, that Bohm's theory is nonlocal) but then define locality a different way when we turn to look at ordinary QM (you know, to make sure we come to the right conclusion, namely, that ordinary QM is local)?

That's sort of like the case with certain sophisticated aether models for electromagnetic interactions. In these models, there is an absolute rest frame, but things conspire to make it impossible to detect this frame.

I agree, there's some similarity there. Although, incidentally, no sophistication is required. If you literally just take Maxwell's equations, assert that there is some one privileged rest frame in which those equations are true, and then explore the consequences of that theory, you will find that already (without any sophisticated or ad hoc additions or corrections) the theory "conspires" to make it impossible to detect which frame is the privileged one. (Bell wrote a nice paper about essentially this point: "how to teach special relativity.")

I think that both an aether model and a Bohm model are aesthetically unappealing, in exactly the same way:they introduce things into the ontology (absolute rest frame, faster-than-light interactions), and then introduce laws of physics that make these things unobservable. That seems like an unnecessary complication---if there is an element of the model that is unobservable, then it seems better to leave it out, if at all possible.

I don't really disagree. But if we're going to be consistent, then we should also diagnose ordinary QM as "aesthetically unappealing" since the wave function is unobservable, and since any rigorous formulation of it requires one to posit an unobservable absolute rest frame (for wave function collapse to happen instantaneously in).

But really this kind of thing should be a discussion for a different day/thread. The important point here is that ordinary QM -- whether one finds it aesthetically appealing or unappealing -- is a nonlocal theory, and so hardly a counter-example to the claim that only nonlocal theories can make the empirically correct predictions. Maybe for now we can just agree to the following: deciding *which* non-local theory provides the best explanation of the correlations is very difficult, depending as it does on squishy things like aesthetic judgments, and indeed perhaps none of the extant options is fully satisfying.

 

[stevendaryl]

Well, I agree (and indeed wrote!) that (again, considering the simple case where a=b) you cannot prove that Alice's measurement influences Bob. There are local ways to explain these perfectly correlations! And there are nonlocal ways to explain them! Who knows which is right. But the point is, Bell's definition of locality gives us a way to assess whether a *particular candidate theory* (say, ordinary QM, or the pilot wave theory, or some "local realist" theory, ...) provides a local explanation or not.

Okay, so you're saying that "there is no local explanation" does not imply "nonlocal"?

 

[ttn]

I agree that if there were probabilities of the form p(a,b,\lambda), then for QM, \lambda would have to be the wave function. However, the space of wave functions is too big to be a probability space in the sense of probability theory. In more rigorous terms, this means that no one has succeeded in specifying a sigma-algebra and a probabilty measure on sufficiently big spaces of wave-functions (like the spaces used in QM, for example L^2(R) or Fock spaces). Thus your p(a,b,\psi)'s can't be probabilities and thus my claim still holds: This definition of locality can't be applied to QM.

That's got to be the strangest argument (for the inapplicability of Bell's formulation of locality to ordinary QM) that I've ever heard. Suffice it to say I disagree. Yes, there are lots and lots of different possible ψs. But I don't think there is any technical problem with this of anything like the sort you suggest here. But rather than get into the details of that, just think about how silly this is. If the space of λs is too big for QM, then surely it's too big for Bohm's theory as well, since the physical states in Bohm's theory include everything they include in QM, plus more stuff. Indeed, each particular ψ corresponds to just one possible physical state in QM, whereas it corresponds to an infinite number of possible physical states in Bohm's theory (since there are an infinite number of different ways the particles could be arranged for that ψ)! So evidently you also think that it is impossible to say whether Bohm's theory is local or not? (I consider that a reductio of your argument.)

You can't make a general formal definition, because different theories have different frameworks that work differently. There is no definition that can cover all possible theories (at least i don't know one).

I'd be interested to talk again after you've actually read about Bell's formulation. It is (roughly) an attempt to do what you say here is impossible. So, look at the attempt, and then tell me how exactly you think it goes wrong. Otherwise it's rather like standing in front of the elephant in the room, with your back to it, explaining how there clearly couldn't possibly be an elephant in the room.

You have to look at your given theory and make up a definition for locality that ensures that your intuitive understanding is fulfilled. In Wightman QFT for example, you postulate that commutators of spacelike separated obervables vanish.

I do agree, actually, that Bell's formulation breaks down for certain "exotic" sorts of theories. But nothing as simple as what you have in mind here. See my paper on "Bell's concept of local causality", which discusses at the end some of the exotic sorts of theories where Bell's formulation starts to run into difficulties.

But if that experiment is performed a hundred times, Bob gets a completely consistent probability distribution that he would also calculate if his particle wouldn't be entangled with Alice's particle.

Correct. In other words, Alice can't send a message to Bob this way. Nevertheless, ordinary QM's explanation of the correlations involves nonlocality.

To make the point more clear: Let's assume every particle of the Earth were entangled with some particle in the andromeda galaxy. Is there a way to find out without traveling to andromeda?

Nope. But nevertheless, according to ordinary QM, if all those particles are entangled in the way you describe, then interventions here can causally influence things going on over there.

 

[stevendaryl]

Now that's an interesting set of statements! So, you agree that, for whatever reason, the Bohm theory precludes signalling (i.e., basically, it agrees with the empirical predictions of QM, including that Bob's marginal shouldn't be affected by Alice's setting). And you want "locality" to just *mean* this no-signalling condition. But then... what in the world do you mean when you say that, despite the no signalling, there ARE nonlocal interactions in the Bohm theory?

Nonlocal in my sense is relative to a set of state variables. Quantum mechanics has no nonlocal interactions in terms of macroscopic variables (the locations, velocities and orientations of macroscopic objects, the values of macroscopic fields). But the Bohm model introduces additional variables (the positions of microscopic particles) that are subject to nonlocal interactions. IF there were some way to know the values of these microscopic variables, then you could signal through them.

So Bohm-type models explain macroscopic variables that have no nonlocal interactions in terms of microscopic variables that do.

 

[nanosiborg]

This idea ... that there is some inconsistency between the theorem and the "experimental design" that makes it improper for us to conclude anything from the experiments -- really makes no sense to me.

Yes, you've made that clear. I'm curious why you put experimental design in quotes.

[ ... snip silly analogy ... ]

Tell me how what you're saying isn't just parallel to that (I think, manifestly absurd) response to the hypothetical scenario.

That should already be clear to you. But, as you've indicated, it isn't. I don't know how to say it any clearer. This isn't your fault. I accept the responsibility for effectively communicating the ideas I'm exploring.

Aspect's experiment (and other more recent and better versions of the same thing) experimentally prove that nature is nonlocal. They falsify locality.

They falsify local theories of quantum entanglement based on Bell's locality condition. They don't prove that nature is nonlocal.

QM is a nonlocal theory, at least by the best definition of locality that we have going -- namely, Bell's as presented in "la nouvelle cuisine". You have a better/different formulation of "locality" to propose? I'm all ears. Or you think there's some flaw in Bell's formulation? I'm all ears.

I'm wondering if you've actually read my posts and thought about the ideas (which are not mine by the way). I've said several times that I agree with you that Bell locality is definitively ruled out as a viable option for modeling quantum entanglement, and that I take Bell's formulation as general. Insofar as effectively exploring the suggestion that QM might be supplemented by LHVs so as to make it a more complete theory of physical reality there's nothing wrong with Bell's formulation. It does that and more, ruling out local theories of quantum entanglement whether HV or realistic or nonrealistic orwhatever.

What it doesn't do is prove that nature is nonlocal.

Quantum teleportation?

There's no physical superluminal transmissions involved in quantum teleportation.

It's clear (to me at least) that you are clinging to loopholes that don't in fact exist, because you don't yet fully appreciate what Bell did.

I'm entertaining and exploring some ideas that I find interesting that you say you don't understand or can't make sense of. They're not 'loopholes' in the usual sense of that word, and I'm not "clinging" to them in the perjorative sense that I take you to mean.

Any clinging that's going on would more appropriately be used to characterize your holding on to the notion that Bell has proved that nature is nonlocal, and your repeated insistence that you simply can't understand or make any sense of the ideas being presented.

So, again, can we just agree to disagree for now? This will be my last post in this thread. You're free to have the last word in our discussion, although I don't see why it would be necessary to reiterate what you've already said unless you want to add some more ad hominems or whatever, as I understand that you can't very well argue (or argue very well) against, or agree with, something that you can't make sense of.

And yes, of course I'll read the papers you suggested. Thanks, sincerely.

 

[rubi]

That's got to be the strangest argument (for the inapplicability of Bell's formulation of locality to ordinary QM) that I've ever heard. Suffice it to say I disagree. Yes, there are lots and lots of different possible ψs. But I don't think there is any technical problem with this of anything like the sort you suggest here.

I'm sorry, but then you are wrong. If you think you are right, then here's a challenge for you: Take the space L^2(\mathbb R) and define some arbitrary example of a probability measure (let's call it \mu) on it (you are absolutely free). Give a meaning to probabilities like for example P(\psi(3) = 5) = \int_{\psi(3)=5}\mathrm d\mu(\psi). I've given you complete freedom here, so if you think that it is possible, this task should be easy. You can provide an arbitary, completely exotic example if you like.

Bell's definition applies only to situations where such a measure is possible. In classical mechanics for example, the space could be \mathbb R^{6N} and the measure could be given by any probability distribution \rho(x_1,p_1,\ldots,x_{3N},p_{3N}) for example.

But rather than get into the details of that, just think about how silly this is. If the space of λs is too big for QM, then surely it's too big for Bohm's theory as well, since the physical states in Bohm's theory include everything they include in QM, plus more stuff. Indeed, each particular ψ corresponds to just one possible physical state in QM, whereas it corresponds to an infinite number of possible physical states in Bohm's theory (since there are an infinite number of different ways the particles could be arranged for that ψ)! So evidently you also think that it is impossible to say whether Bohm's theory is local or not? (I consider that a reductio of your argument.)

Bohms theory isn't nonlocal with resprect to Bell's definition (because it can't be applied) but in the sense of whether there is an action at a distance or not.

 

[ttn]

Yes, you've made that clear. I'm curious why you put experimental design in quotes.

Because it seems like what you actually mean is the *results of*, rather than the *design of*, the experiments. But this is just another way of saying I don't understand what you're getting at, and as you suggest below, it is perfectly reasonable to just leave it there for now.

They falsify local theories of quantum entanglement based on Bell's locality condition. They don't prove that nature is nonlocal.

I would put the first sentence this way: they falsify local theories of quantum entanglement, where "local" is defined in the way Bell defined it.

If we agree about that (and I'm honestly not sure), then the second sentence should read: "This *does* prove that nature is nonlocal (with "local" defined in Bell's way)."

I know you said you didn't want to post more, and trust me, I respect and understand that -- but what I was never able to understand was whether you were saying (a) that Bell's def'n of locality was fine, but that there was some subtle logical presupposition in the analysis *other* than Bell's def'n of locality, or (b) that there is some problem/flaw in Bell's def'n. So, maybe that expression of my confusion will help you sort out how to communicate your idea more effectively next time. Or possibly it's just that I'm dense.

I'm wondering if you've actually read my posts and thought about the ideas (which are not mine by the way). I've said several times that I agree with you that Bell locality is definitively ruled out as a viable option for modeling quantum entanglement, and that I take Bell's formulation as general.

Yes, I've read every word. And I've heard you say those things. But then I hear you saying "but..." with the "..." being stuff that, to me, contradicts the above. So I am continuously thinking that either you must not have meant what you said, or I didn't understand it correctly.

There's no physical superluminal transmissions involved in quantum teleportation.

There's no transmission of useable *information*, to be sure. That is, you can't transmit a *message* superluminally this way. But you'd be hard pressed to explain the fact that quantum teleportation is possible, in terms of a local theory.

So, again, can we just agree to disagree for now? This will be my last post in this thread. You're free to have the last word in our discussion, although I don't see why it would be necessary to reiterate what you've already said unless you want to add some more ad hominems or whatever, as I understand that you can't very well argue (or argue very well) against something that you can't make sense of.

Of course. I'm sincerely sorry if my posts have come off as attacking you. That wasn't intended at all. I was just trying (perhaps too hard?) to understand what you were saying. And the reason I kept going back to the general points about Bell's theorem is not that I ignored your statements about where you agreed with me -- rather I was just trying to keep this part of the thread it's embedded in, i.e., connect it back, largely for the purposes of other people who might be reading, to the big issue at hand here, namely, whether Bell's theorem should be understood as refuting "realism" or "locality". Sorry if the attempt to keep both of those balls in the air (talking with you and arguing for a general audience about the main issue of the thread) made it seem like I was throwing balls at you undeservedly.

 

[nanosiborg]

T. Norsen, sorry if I came off as having taken offence. I actually enjoyed much of our discussion, and will continue to enjoy the other discussions in this thread from the sidelines. Thanks for clarifying, and I realize that it's up to me to put into clearly understandable form any ideas that I might want help in exploring. Of course, that's part of the problem I'm having, as I just have this vague intuitive notion that there might be something there, but am not sure how to state it most clearly. Maybe after reading the papers you suggested I won't have to worry about that.

 

[audioloop]

Interesting question.

So, which is it? Actually both are true! The key point here is that, according to the pilot-wave theory, there will be many physically different ways of "measuring the same property". Here is the classic example that goes back to David Albert's classic book, "QM and Experience." Imagine a spin-1/2 particle whose wave function is in the "spin up along x" spin eigenstate. Now let's measure its spin along z. The point is, there are various ways of doing that. First, we might use a set of SG magnets that produce a field like B_z ~ B_0 + bz (i.e., a field in the +z direction that increases in the +z direction). Then it happens that if the particle starts in the upper half of its wave packet (upper here meaning w.r.t. the z-direction) it will come out the upper output port and be counted as "spin up along z"; whereas if it happens instead to start in the lower half of the wave packet it will come out the lower port and be counted as "spin down along z". So far so good. But notice that we could also have "measured the z-spin" using a SG device with fields like B_z ~ B_0 - bz (i.e., a field in the z-direction that *decreases* in the +z direction). Now, if the particle starts in the upper half of the packet it'll still come out of the upper port... *but now we'll call this "spin down along z"*. Whereas if it instead starts in the lower half of the packet it'll still come out of the lower port, but we'll now call this *spin up along z*.

And if you follow that, you can see the point. Despite being fully deterministic, what the outcome of a "measurement of the z-spin" will be -- for the same exact initial state of the particle (including the "hidden variable"!) -- is not fixed. It depends on which *way* the measurement is carried out!

i agree.
I can think of another example, "position" position respect to ?

.

 

[stevendaryl]

And if you follow that, you can see the point. Despite being fully deterministic, what the outcome of a "measurement of the z-spin" will be -- for the same exact initial state of the particle (including the "hidden variable"!) -- is not fixed. It depends on which *way* the measurement is carried out!

This point, which I think I knew once upon a time, but forgot, is very interesting. It bears some similarity with Cramer's "Transactional Interpretation". In that interpretation, the result of a measurement was not completely random, but could depend on details in the future. The transactional interpretation is sort of nonlocal, as well, but the nonlocal interactions propagate along null paths into the future and into the past. Maybe the two theories end up being essentially the same?

 

[ttn]

I'm sorry, but then you are wrong. If you think you are right, then here's a challenge for you: Take the space L^2(\mathbb R) and define some arbitrary example of a probability measure (let's call it \mu) on it (you are absolutely free). Give a meaning to probabilities like for example P(\psi(3) = 5) = \int_{\psi(3)=5}\mathrm d\mu(\psi). I've given you complete freedom here, so if you think that it is possible, this task should be easy. You can provide an arbitary, completely exotic example if you like.

This really isn't the place for a big technical discussion of this kind of thing. Suffice it to say that what you are saying here is totally irrelevant to Bell's formulation. Go read up on how he defines "locality" and you'll see that nothing like this comes up.

Bohms theory isn't nonlocal with resprect to Bell's definition (because it can't be applied) but in the sense of whether there is an action at a distance or not.

But it is precisely "the sense of whether there is an action at a distance or not" that Bell is concerned with, and that his definition captures. You should look into how he defines this idea, before you decide whether it's applicable to Bohm's (or some other) theory and before you decide whether or not it genuinely captures the notion of "no action at a distance".

 

[stevendaryl]

This really isn't the place for a big technical discussion of this kind of thing. Suffice it to say that what you are saying here is totally irrelevant to Bell's formulation. Go read up on how he defines "locality" and you'll see that nothing like this comes up.

I think it does. Bell assumed a probability distribution on the "hidden variable" \lambda. So technically, if the "hidden variable" is a function, with infinitely many degrees of freedom, then there can't be a probability distribution.

This technicality was exploited by Pitowsky, who developed a local hidden variables theory that makes the same predictions for the spin-1/2 EPR experiment as orthodox quantum mechanics. Where he escapes from Bell's clutches is exactly in using a "hidden variable" for which there is no probability distribution. He uses nonmeasurable sets, constructed via the continuum hypothesis.

 

[ttn]

i agree.
I can think of another example, "position" position respect to ?

.

Hmmm. Maybe I'm not entirely sure what you are intending to give another example of, but it is actually not true that position is "contextual" (in the way I explained spin was) for Bohm's theory. For position measurements (only!) there is, in Bohm's theory, a definite unambiguous pre-existing value (namely, the actual location of the thing in question) that is simply passively revealed by the experiment.

That's probably not what you meant. You meant something about the arbitrariness of reference frame -- e.g., what you call x=5, maybe I call x=-17. But that's a totally different issue than the one I was bringing up for spin in bohm's theory. There is an analog of your issue for spin -- namely, maybe what you call "spin along z = +1" I instead call "spin along z = +hbar/2" or "spin along z = 37". All of those, actually, are perfectly valid choices. We can disagree about what to *call* a certain definite outcome. But that is not at all the point of the example I explained for the contextuality of spin in bohm's theory. There, the point is not that different people might call the outcome different things, but that two different experiments (that happen to correspond to the same Hermitian operator in QM) can yield distinct outcomes (for exactly the same input). This isn't about calling the same one outcome by two different names; the outcomes are really genuinely distinct.

 

[audioloop]

Hmmm. Maybe I'm not entirely sure what you are intending to give another example of, but it is actually not true that position is "contextual" (in the way I explained spin was) for Bohm's theory. For position measurements (only!) there is, in Bohm's theory, a definite unambiguous pre-existing value (namely, the actual location of the thing in question) that is simply passively revealed by the experiment.

That's probably not what you meant. You meant something about the arbitrariness of reference frame -- e.g., what you call x=5, maybe I call x=-17. But that's a totally different issue than the one I was bringing up for spin in bohm's theory. There is an analog of your issue for spin -- namely, maybe what you call "spin along z = +1" I instead call "spin along z = +hbar/2" or "spin along z = 37". All of those, actually, are perfectly valid choices. We can disagree about what to *call* a certain definite outcome. But that is not at all the point of the example I explained for the contextuality of spin in bohm's theory. There, the point is not that different people might call the outcome different things, but that two different experiments (that happen to correspond to the same Hermitian operator in QM) can yield distinct outcomes (for exactly the same input). This isn't about calling the same one outcome by two different names; the outcomes are really genuinely distinct.

i understand, but what is a definite value ? something defined by other definite value in turn defined by another value and so on.
in the case of position x,y,z axes in turn determined by other set of axes ? in turn determined by other set of axes ?

"coordinates" respect to ?

 

[rubi]

This really isn't the place for a big technical discussion of this kind of thing. Suffice it to say that what you are saying here is totally irrelevant to Bell's formulation. Go read up on how he defines "locality" and you'll see that nothing like this comes up.

As far as I'm concerned, his definition of locality requires the existence of the probabilities of the form p(a,b,\lambda), so if they don't exist (which definitely is the case in QM even for the simple case of a free 1D particle), then the definition can't be applied. Up to now, p(a,b,\psi) is only a purely formal expression void of any precise meaning. In particular, it's not a probability.

By the way: I was looking for that paper you suggested, but i don't find it on the internet. (Apart from that, i don't know french, so i probably couldn't read it?) Can you point me to a source? I have access to most journals.

But it is precisely "the sense of whether there is an action at a distance or not" that Bell is concerned with, and that his definition captures. You should look into how he defines this idea, before you decide whether it's applicable to Bohm's (or some other) theory and before you decide whether or not it genuinely captures the notion of "no action at a distance".

I'd like to like to look into this, but as i said: I don't find that paper anywhere. However, if it uses probabilities of the form p(a,b,\lambda), then it's not applicable.

 

[ttn]

I think it does. Bell assumed a probability distribution on the "hidden variable" \lambda.

Not in the definition of locality! (Yes, such a thing does come up in the derivation of the inequality, though.)

This technicality was exploited by Pitowsky, who developed a local hidden variables theory that makes the same predictions for the spin-1/2 EPR experiment as orthodox quantum mechanics. Where he escapes from Bell's clutches is exactly in using a "hidden variable" for which there is no probability distribution. He uses nonmeasurable sets, constructed via the continuum hypothesis.

I am highly skeptical of this. First of all, the claim that was made here was that Bell's definition of locality is inapplicable if the space of λs is unmeasureable. That is simply false, and the person making such a claim obviously hasn't actually read/digested Bell's formulation of locality. (Probably anybody making this claim simply doesn't yet appreciate that there's a difference between Bell's definition of locality, and Bell's inequality.) But anyway, were it true, then wouldn't it follow that it was impossible to meaningfully assert that Pitowsky's model is local? Yet that is asserted here. So something is amiss. Furthermore, if the space of λs is unmeasureable, I don't see how you could possibly claim that the theory "makes the same predictions ... as orthodox quantum mechanics".

I'd even be willing to bet real money that this isn't right -- that is, that there's no genuine example of a local theory sharing QM's predictions here. If it were true, it would indeed be big news, since it would refute Bell's theorem! (Something that many many people have wrongly claimed to do, incidentally...) But internet bets don't usually end well -- more precisely, they don't usually end at all, because nobody will ever concede that they were wrong. So instead I'll just say this: you provide a link to the paper, and I'll try to find time to take a look at it and find the mistake.

 

[ttn]

i understand, but what is a definite value ? something defined by other definite value in turn defined by another value and so on.
in the case of position x,y,z axes in turn determined by other set of axes ? in turn determined by other set of axes ?

"coordinates" respect to ?

I don't think there's any serious issue here that has any relevance to Bell's theorem. Surely it is possible to specify a coordinate system in such a way that different people can adopt and use that same system and thus communicate unambiguously with each other about exactly where some pointer (indicating the outcome of an arbitrary measurement) is.