Copenhagen: Restriction on knowledge or restriction on ontology?

[DarMM]

But is the sample space ##p\left(b_1\right)## from the first experiment (##A_1, B_1##) the same as sample space ##p\left(b_1\right)## from the second experiment (##A_2, B_1##)?

The set of ##b_1## outcomes isn't a sample space for any of the measurement choices. Pairs like ##(a_1,b_1)## are.

 

[zonde]

The set of ##b_1## outcomes isn't a sample space for any of the measurement choices. Pairs like ##(a_1,b_1)## are.

This does not sound right. Surely I can look at Bob's measurement results without looking at Alice's results.

 

[DarMM]

This does not sound right. Surely I can look at Bob's measurement results without looking at Alice's results.

Well if you only consider Bob's outcome and never Alice's it's not a CHSH test.

However I think your question might really be more about the marginals for a Bob-solo test as constructed from the CHSH predictions. If you look at just Bob's test the probability distribution is then simply a marginal of either the probability distributions ##p(a_1,b_1)## or ##p(a_2,b_1)##. The marginalisation results in the same distribution for ##b_1## regardless as is consistent with no signalling.

 

[zonde]

Perhaps this will work better. Say Alice and Bob perform billions of experiments on observables ##A_1, A_2, B_1, B_2##. Each time they perform one of four pair measurements:

1. ##A_1, B_1##
2. ##A_2, B_1##
3. ##A_1, B_2##
4. ##A_2, B_2##

The point is that afterwards when they look at the statistics they are incompatible with the notion that the results are drawn from a distribution ##p\left(a_1,a_2,b_1,b_2\right)## over a common sample space. They can only be drawn from individual distributions like ##p\left(a_1,b_1\right)##.

Thus the approach these views take is that is because in a ##A_1, B_1## test only those two variables gain values the sample space is just the space of pairs ##\left(a_1,b_1\right)##. Since the other variables have no value in a ##A_1, B_1## test, not unmeasured values but none, the distribution ##p\left(a_1,b_1\right)## has more freedom because it is not a marginal of some ##p\left(a_1,a_2,b_1,b_2\right)##.

In short the measurements Alice and Bob perform need not be consistent with another they could have performed.

If you talk only about ##A_1, B_1## experiment it's not CHSH experiment. CHSH requires all four experiments.
So we unify all four experiments in one experiment and sample space for that experiment is ##\{a_i,b_i\}##.

But we have to take into account locality condition. Locality condition is that whatever happens in one laboratory does not affect spacelike separated event in other laboratory i.e. for any event in laboratory A alternative events are possible in laboratory B for any value of independent test parameter.
This locality condition means that it should be possible to establish certain relationship within this sample space. For any event ##a_i## in laboratory A there should be subset of possible events for any value of independent parameter in laboratory B (subsets ##\{b_1\}## and ##\{b_2\}##). In other words there should be common values of ##a_i## in sample sets ##\{a_i,b_1\}## and ##\{a_i,b_2\}##. And the same goes the other way around.
What this means is that it should be possible to form sample set ##\{a_1,a_2,b_1,b_2\}## from ##\{a_i,b_i\}## if the model is local.

 

[bahamagreen]

When Fourier analyzes an arbitrary waveform into sine waves, it looks like the choice of using a sine wave sets an experimental condition for measurement, and it "summons" the measurement result from the arbitrary wave being measured. That measurement result describes some property or attribute of the arbitrary wave.

Fourier works using other choices of basis "measurement" waves as well... cosine, triangle, square, impulse, but also (with much more difficulty) any arbitrary waveform. Don't the simplest measuring waveforms correspond to relatively simple experimental conditions and familiar attributes, like momentum or position?

Fourier works using any arbitrary waveform as the basis for decomposition, although the experimental condition that it describes might be incomprehensibly complex and technologically impossible to arrange, and even if possible the resulting measured attribute might be incomprehensible.

But if this is so, (if you understand me in spite of my poor corrupted description, without rigor), it looks like all possible properties or attributes in principal may be measured, depending on which basis is chosen. The questions of whether the attributes exist prior to measurement or are mathematically describable seem ill-formed to me with respect to this comparison to Fourier.

Is there something fundamentally wrong or missing with the Fourier basis "interpretation"?

 

[DarMM]

If you talk only about ##A_1, B_1## experiment it's not CHSH experiment. CHSH requires all four experiments.

Of course, but I never spoke about only testing that pair. I list all four. I was just responding to your question about ##b_1## which takes place in those pairs.

So we unify all four experiments in one experiment and sample space for that experiment is ##\{a_i,b_i\}##.

The point is that the statistics of those experiments are incompatible with each pair being drawn from the same sample space.

 

[stevendaryl]

The point is that the statistics of those experiments are incompatible with each pair being drawn from the same sample space.

I'm a little irritated by the phrase "not drawn from the same sample space". I sort of know what it means, but to the extent that it is meaningful, it just seems to be a restatement of the facts of Bell-violating statistics, rather than an explanation of them.

A strange fact of quantum mechanics, which is illustrated by Bell-violating experiments, is that it does not provide a way to calculate probabilities until after a choice of measurement is made. It doesn't say "An electron produced by such-and-such procedure has such-and-such probability of BEING spin-up in the z-direction", it says "the electron has such-and-such probability of being measured to have spin-up, if observable being measured is the z-component of spin". This relates to Bell's term "beable" versus "observable". Quantum mechanics gives probabilities for what is observed, but not for what is true.

With quantum mechanics, it's as if nature waits until you choose what you want to measure, then rolls the dice to figure out what happens. But in an EPR-type experiment, the decision of what measurement to be performed can be made at the last moment, when it's seemingly too late to roll the dice.

 

[DarMM]

I'm a little irritated by the phrase "not drawn from the same sample space". I sort of know what it means, but to the extent that it is meaningful, it just seems to be a restatement of the facts of Bell-violating statistics, rather than an explanation of them.

These are non-representational views so naturally they don't provide an explanation or claim to right?

All they're saying is that the absense of counterfactuals (i.e. no common sample space) is what permits these correlations and locality.

As for why nature lacks counterfactuals, that goes unexplained. In fact Bohr went as far as thinking the lack of counterfactuals (in his terminology "complimentarity") blocks human comprehension of the microscopic world. This is essentially where the incompleteness you often mention shows up in these views.

 

[stevendaryl]

These are non-representational views so naturally they don't provide an explanation or claim to right?

My point is that it doesn't seem to clarify anything.

All they're saying is that the absense of counterfactuals (i.e. no common sample space) is what permits these correlations and locality.

I don't think it does that. If it did, that would count as an explanation in my mind.

 

[DarMM]

I don't think it does that. If it did, that would count as an explanation in my mind.

Can you explain why? Perhaps these views have no answer to your issue, but I'm not sure what the issue is. What does the absence of counterfactuals fail to explain regarding these correlations and locality?

 

[stevendaryl]

Can you explain why? Perhaps these views have no answer to your issue, but I'm not sure what the issue is. What does the absence of counterfactuals fail to explain regarding these correlations and locality?

My feeling is that talking about "counterfactuals" and "different sample spaces" does not contribute anything to the discussion of whether QM is nonlocal, what Bell's inequality tells us about reality, etc. It's just words offered in place of understanding. My eyes glaze over when people start using these words. It's not that I don't understand them, but that they seem vacuous.

 

[DarMM]

I don't understand sorry, that counterfactuals don't exist seems like a clear statement to me. As clear as retrocausality and nonlocal influences at any rate. What's the issue?

 

[stevendaryl]

I don't understand sorry, that counterfactuals don't exist seems like a clear statement to me. As clear as retrocausality and nonlocal influences at any rate. What's the issue?

It's clear enough, but it seems to me just a restatement of the facts that serves no purpose in advancing understanding. It gives the illusion of explanation.

 

[bahamagreen]

When Fourier analyzes an arbitrary waveform into sine waves, it looks like the choice of using a sine wave describes an experimental condition for measurement, and it "summons" the measurement result from the arbitrary wave being measured. That measurement result describes some attribute of the arbitrary wave.

Fourier works using other choices of "measurement" waves as well... cosine, triangle, square, impulse, but also (with much more difficulty) any arbitrary waveform. Don't the simplest measuring waveforms correspond to relatively simple experimental conditions and familiar attributes, like momentum or position?

Fourier works using any arbitrary waveform as the basis for decomposition, although the experimental condition that it describes might be incomprehensibly complex and technologically impossible to arrange, but even if possible the resulting measured attribute might be incomprehensible.

But if this is so, (my poor corrupted description, without rigor), it looks like one must take the view that all possible properties or attributes in principal may be measured, depending on which basis is chosen. The question of whether the thing being measured has those properties or attributes prior to measurement or whether they are mathematically describable seems like an ill-formed question to me with respect to this comparison to Fourier.

Is this Fourier "interpretation" flawed, or missing something?

 

[DarMM]

It's clear enough, but it seems to me just a restatement of the facts that serves no purpose in advancing understanding. It gives the illusion of explanation.

Okay. The way I would have seen it, counterfactual indefinitness is a novel feature of the logic of the world. You disagree with this in some form? What about its productive use in resolving post and pre-selection paradoxes and in quantum information?

Many Worlds also has counterfactual indefiniteness, do you find it unsatisfying there as well?

 

[stevendaryl]

Okay. The way I would have seen it, counterfactual indefinitness is a novel feature of the logic of the world.

I don't see that. If there is randomness in the world, then it seems to me that the world would lack counterfactual definiteness. If flipping a coin randomly results in "heads" or "tails", then there is no answer to the question: "What would the result have been if I had flipped the coin?" If you didn't flip the coin, then there is no definite answer to the question of what you would have gotten.

That doesn't give any insight into quantum mechanics, because classical stochastic theories lack counterfactual definiteness, as well.

The strangeness of quantum mechanics is not about the nondeterminism, but about the certainty. If Alice gets spin-up for a measurement of the z-component of spin for her particle, then it is certain that Bob will get spin-down for a measurement of the z-component of spin for his particle. It's not the lack of definiteness that is interesting, it is the certainty. Or rather, the combination of certainty and randomness.

So saying that "quantum mechanics lacks counterfactual definiteness" does not, in any way, get at the heart of what is strange about quantum mechanics.

 

[DarMM]

That doesn't give any insight into quantum mechanics, because classical stochastic theories lack counterfactual definiteness, as well.

I'm having a hard time understanding this, unless you mean something different by the phrase, as all Kolmogorov stochastic processes have counterfactual definitness as far as I understand the mathematics.

 

[stevendaryl]

I'm having a hard time understanding this, unless you mean something different by the phrase, as all Kolmogorov stochastic processes have counterfactual definitness as far as I understand the mathematics.

Counterfactual definiteness means that there is a definite answer to counterfactual questions: "What would have happened if I had done X rather than Y?" What do you mean by "counterfactual definiteness"?

 

[vanhees71]

The title of this thread is motivated by frequent arguments I had with other members here, especially @DarMM and @vanhees71 .

The so called "Copenhagen" interpretation of QM, known also as "standard" or "orthodox" interpretation, which is really a wide class of related but different interpretations, is often formulated as a statement that some things cannot be known. For instance, one cannot know both position and momentum of the particle at the same time. But on other hand, it is also not rare that one formulates such an interpretation as a statement that some things don't exist. For instance, position and momentum of the particle don't exist at the same time.

Which of those two formulations better describes the spirit of Copenhagen/standard/orthodox interpretations? To be sure, adherents of such interpretations often say that those restrictions refer to knowledge, without saying explicitly that those restrictions refer also to existence (ontology). Moreover, some of them say explicitly that things do exist even when we don't know it. But in my opinion, those who say so are often inconsistent with other things they say. In particular, they typically say that Nature is local despite the Bell theorem, which is inconsistent. It is inconsistent because the Bell theorem says that if something (ontology, reality, or whatever one calls it) exists, then this thing that exists obeys non-local laws. So one cannot avoid non-locality by saying that something is not known. Non-locality implied by the Bell theorem can only be avoided by assuming that something doesn't exist. Hence any version of Copenhagen/standard/orthodox interpretation that insists that Nature is local must insist that this interpretation puts a severe restriction on the existence of something, and not merely on the possibility to know something.

Well, much confusion is due to unsharp formulations. E.g., the most successful physical theory ever, the Standard Model of elementary particle physics, is indeed a local relativistic QFT. That's a very short characterization of a theory you need at least 4-5 semesters of careful studies in theoretical physics to formulate it.

What's local are the interactions, i.e., the interactions are described by a sum over monomials of field operators at one space-time point, where the field operators fulfill the microcausality condition.

What's "non-local" in the context of EPR, Bell, etc. are correlations between parts of a quantum system. One of the most simple examples and nowadays also most simply prepared thanks to the development in laser technology are polarization-entangled two-photon states (prepared through parametric downconversion, a non-linear-quantum-optics effect). The two photons are produced according to local (!) interactions of laser light with a birefringent crystal, with quite well definied different momenta. Afer a while the probability to detect both photons is peaked at far-distant positions. The polarizations are entangled, e.g., in the singlet state ##(|HV\rangle -|VH \rangle)/\sqrt{2}##. The far-distant measurements lead to 100% correlations of the polarization state, i.e., if A measures horizontal polarization for her photon, B necessarily finds vertical polarization and vice versa. These measurements can take place at as much distance as you like, provided none of the photons are interacting with anything destroying this "non-local" correlation of the polarizations of these two photons. These correlations are 100% although the single-photon polarizations are maximal uncertain in this case, i.e., A and B get truly unpolarized photons.

In my opinion the use of the notion of "non-locality" is misleading. I prefer to call it long-range correlations to make the notion distinct from the notion of "locality" in the very important fundamental sense as "local interactions" in a "local"relativistic QFT.

 

[zonde]

The point is that the statistics of those experiments are incompatible with each pair being drawn from the same sample space.

If you have on mind sample space ##\{a_1,a_2,b_1,b_2\}## when you say this then it is of course true.
But you quoted me referring to sample space ##\{a_i,b_i\}##. Obviously statistics are compatible with this sample space as it is statement of experimental facts. It just unifies sample spaces of individual experiments without imposing any restrictions on these subspaces.

 

[DarMM]

If you have on mind sample space ##\{a_1,a_2,b_1,b_2\}## when you say this then it is of course true.
But you quoted me referring to sample space ##\{a_i,b_i\}##. Obviously statistics are compatible with this sample space as it is statement of experimental facts. It just unifies sample spaces of individual experiments without imposing any restrictions on these subspaces.

Does the ##i## in your sample space refer to the iteration of the experiment?

 

[zonde]

Does the ##i## in your sample space refer to the iteration of the experiment?

No, it refers to measurement settings as in your post #33.

 

[DarMM]

No, it refers to measurement settings as in your post #33.

How is it different from the four variable sample space?

 

[DarMM]

Counterfactual definiteness means that there is a definite answer to counterfactual questions: "What would have happened if I had done X rather than Y?" What do you mean by "counterfactual definiteness"?

That's not the typical meaning in quantum foundations, probability or information theory. On the phone so latex is a bit difficult, I'll make a longer post later.

 

[zonde]

Ok, I should have written ##\{a_i,b_j\}##. My mistake.

How is it different from the four variable sample space?

It contains only pairs of outcomes.
There is not much of an argument attached to this sample space itself, it is just used for convenient notation.

 

[DarMM]

Ok, I should have written ##\{a_i,b_j\}##. My mistake.

It contains only pairs of outcomes.
There is not much of an argument attached to this sample space itself, it is just used for convenient notation.

Is that not just a cardinality four collection of two variable sample spaces?

 

[stevendaryl]

That's not the typical meaning in quantum foundations, probability or information theory. On the phone so latex is a bit difficult, I'll make a longer post later.

My feeling is that if it doesn't mean that, then it's badly named. This paper gives a definition:
https://arxiv.org/pdf/1605.04889.pdf

A counterfactually definite theory is described by a function (or functions) that map(s) tests onto
numbers. The variables of the function(s) argument(s) must be chosen in a one to one correspondence to physical entities that describe the test(s) and must be independent variables in the sense
that they can be arbitrarily chosen from their respective domains.

This definition means that the outcomes of measurements must be described by functions of a set of independent variables.

That seems in keeping with what I said: If it's counterfactually definite, then it means that the outcomes of measurements are functions of the variables describing the state of the system (plus measuring device) at the time the measurement is performed. That is not the case with a stochastic theory. In a stochastic theory, the outcome of a measurement is not determined (so it's not a function of any particular collection of variables).

 

[zonde]

Is that not just a cardinality four collection of two variable sample spaces?

Yes, if I understand you correctly.

 

[DarMM]

Yes, if I understand you correctly.

Okay, so how is that different from my description?

 

[zonde]

Okay, so how is that different from my description?

Well, it is one sample space, not four.

 

[DarMM]

Well, it is one sample space, not four.

Okay I'm very confused now. Is it the union of my sample spaces?

 

[zonde]

Okay I'm very confused now. Is it the union of my sample spaces?

Yes, of course. What's so confusing about it?

 

[DarMM]

Yes, of course. What's so confusing about it?

The fact that the union isn't a sample space. Can you give it a measure ##\mu## with ##\mu(\Omega) = 1##? With ##\Omega## the union you're describing.

 

[zonde]

This is the sample space:
##\{(h_1, h_1), (v_1, v_1), (h_1, v_1), (v_1, h_1),##
##(h_1, h_2), (v_1, v_2), (h_1, v_2), (v_1, h_2),##
##(h_2, h_1), (v_2, v_1), (h_2, v_1), (v_2, h_1),##
##(h_2, h_2), (v_2, v_2), (h_2, v_2), (v_2, h_2)\}##
"h" or "v" stands for two different outcomes and indexes 1 and 2 for two different measurement settings.

 

[DarMM]

That's perfect, but in order for it to be a sample space what's the measure?