Entanglement swapping, monogamy, and realism

[PeterDonis]

There are no correlations between 2&3 or 1&4 in the initial state. I haven't claimed anything like that.

Yes, you did, in what I already quoted from you:

the observed entanglement of 1&4 in the ensemble selected through the BSM on 2&3 is due to the correlations described by the entanglement of the photon pairs 1&2 and 3&4 in the initial state

This can only explain entanglement between 1 & 4 if there are correlations between 1 & 4 in the initial state. But now you say (correctly) that there aren't.

 

[vanhees71]

Sigh. What I wanted, of course to say, is that the photons in pair (12) and the photons in pair (34) are entangled. One should never use words. Please see #105 for clarification.

 

[PeterDonis]

What I wanted, of course to say, is that the photons in pair (12) and the photons in pair (34) are entangled.

Yes, I agree with that, in the initial state. But what about the final state, after the BSM is done?

 

[vanhees71]

Yes, I agree with that, in the initial state. But what about the final state, after the BSM is done?

After you project (23) in one of the four Bell states, (14) must necessarily be in the same Bell state (see #105). I think nobody disagrees with this.

The point, however, is that this is not due to some "spooky interaction at a distance" between the spacelike separated BSM projection measurement on (23) and photons 1 and 4 but simply due to the correlations due to entanglement of pair (12) and the entanglement of (34) in the initial state (of course (14) and (23) are not entangled in the initial state).

 

[PeterDonis]

After you project (23) in one of the four Bell states, (14) must necessarily be in the same Bell state (see #105). I think nobody disagrees with this.

Nobody disagrees with the math, but I think there is considerable disagreement on how to interpret it.

simply due to the correlations due to entanglement of pair (12) and the entanglement of (34) in the initial state

But how does this work? The math doesn't say. If you are satisfied with just pointing at the math and leaving any question the math doesn't answer unanswered, that's fine. But many people aren't; that's why this thread exists. And it doesn't seem like that's the position you're taking anyway: you're asserting what I quoted above as an explanation; you're not saying that no explanation is required at all.

On its face, it seems like you are describing a local hidden variable model of the kind that is ruled out by Bell's Theorem.

 

[DrChinese]

Consider a pair of spacelike separated entangled particles in the state $|\uparrow_A\uparrow_B\rangle + |\downarrow_A\downarrow_B\rangle$. You have particle $A$ and I have $B$. You measure $A$ in the $\{\uparrow, \downarrow\}$ basis and obtain the outcome $\uparrow$, so you know that if I measure $B$, I will obtain the outcome $\uparrow$. My questions to you:

i) If, for the same experimental run (i.e. not merely a new/different run) you had instead not performed a measurement, would I still have been guaranteed to obtain the outcome $\uparrow$?

ii) Is your answer to i) interpretantion-dependent, or enforced by the formalism of QM? Or can QM even concern such questions that are not strictly resolvable by experiment?

I would agree with @vanhees71 :

i) No, because all you know for the full ensemble is that your particle is ideally unpolarized. You simply get in 50% of the cases up and in 50% of the cases down (in the usual statistical sense).

DrC: You must look at the full initial and final contexts. The general rule is not to ascribe a specific value to an unmeasured observable.

ii) It may be interpretation dependent. ...

There are interpretations that answer i. differently. A true realistic interpretation would say there is a specific value independent of the act of measurement, even if it is unknowable in principle.

 

[kurt101]

I'm not sure either Bell or Einstein made the same claims as you are making. That is why you should give a specific reference.

What I meant is that Bell or Einstein, used the phrasing of action at a distance in their papers. For example Bell uses it in his BERTLMANN'S SOCKS AND THE NATURE OF REALITY paper: https://hal.science/jpa-00220688/document

If you think either Einstein or Bell made this claim, please give a specific reference.

I don't.

On its face, what you describe here and in the rest of your post looks like a local hidden variable interpretation of the sort that is ruled out by Bell's Theorem.

I am NOT describing a local hidden variable theory. The action that I am using is the spooky action at a distance like Bell uses in his BERTLMANN'S SOCKS AND THE NATURE OF REALITY paper: https://hal.science/jpa-00220688/document.

 

[PeterDonis]

What I meant is that Bell or Einstein, used the phrasing of action at a distance in their papers.

That's just a phrasing to describe the issue. It doesn't describe any particular interpretation.

I am NOT describing a local hidden variable theory.

Then I have no idea what "realist" interpretation you think you are using or what you think it says. All you are doing is describing the issue under discussion. That is pointless; the discussion is not about describing the issue, it's about what possible resolutions of it might be.

 

[DrChinese]

1. After you project (23) in one of the four Bell states, (14) must necessarily be in the same Bell state (see #105). I think nobody disagrees with this.

2. The point, however, is that this is ... simply due to the correlations due to entanglement of pair (12) and the entanglement of (34) in the initial state (of course (14) and (23) are not entangled in the initial state).

1. Agreed by me.

2. What does this even mean? What correlations? You agree that (14) are NOT entangled in the initial state (good, we agree). And all of them (in the ideal case) are entangled in one of the 4 Bell states in the final state (even if that particular state is unknown).

I have said this before: Certainly there is nothing that connects the successful swap to any change in the statistical relationship of the (14) pairs in *your* reading of the experiment. But we are talking about perfect correlations and violation of Bell inequalities here. According to your reading, EVERY entangled pair anywhere in the universe at any time could be shown to have the same Bell correlations with (12) or (34) - and each other. Because in your mind, those correlations are present (pre-existing) in any 2 entangled pairs - they are just waiting to be uncovered. (Since of course, nothing actual happens/changes with a BSM - in your worldview).

How do you make any physical sense of that? Any 2 entangled pairs anywhere have these hidden correlations, waiting to be revealed? Are you really saying that?

 

[mattt]

1. Agreed by me.

2. What does this even mean? What correlations? You agree that (14) are NOT entangled in the initial state (good, we agree). And all of them (in the ideal case) are entangled in one of the 4 Bell states in the final state (even if that particular state is unknown).

I have said this before: Certainly there is nothing that connects the successful swap to any change in the statistical relationship of the (14) pairs in *your* reading of the experiment. But we are talking about perfect correlations and violation of Bell inequalities here. According to your reading, EVERY entangled pair anywhere in the universe at any time could be shown to have the same Bell correlations with (12) or (34) - and each other. Because in your mind, those correlations are present (pre-existing) in any 2 entangled pairs - they are just waiting to be uncovered. (Since of course, nothing actual happens/changes with a BSM - in your worldview).

How do you make any physical sense of that? Any 2 entangled pairs anywhere have these hidden correlations, waiting to be revealed? Are you really saying that?

Yes, he will accept that (because the mathematical analysis will be identical for (1,2)×(3,4) than for (1,2)×(5,6) or any other entangled pair ( (7,8) , (9,10), ... ) created independently anywhere, anytime.

There will be maximally entangled subensembles (1,4) ( or (1,6), or (1,8).... depending on what projection to a BSM State we make with what pairs (2,3) (or (2,5), or (2,7)....) respectively.

He just accepts it as a given (because the mathematics of QM says that) and I think he doesn't need any "more profound" explanation.

In my case, I'm still thinking about it...

 

[kurt101]

Then I have no idea what "realist" interpretation you think you are using or what you think it says. All you are doing is describing the issue under discussion. That is pointless; the discussion is not about describing the issue, it's about what possible resolutions of it might be.

I am trying to get closure on this statement from @DrChinese who said my previous statement "Fundamentally coincidence is the driver factor that allows 1 & 4 to be entangled" doesn't make sense because BSM is the driving factor. I am trying to clarify my statement and I want to know if we understand each other.

Yours is not what I would call a realistic interpretation. "Fundamentally coincidence is the driving factor that allows 1 & 4 to be entangled" doesn't make sense, because the BSM is the driving factor. Each and every BSM results in 1 & 4 entanglement, and that does not occur otherwise.

I am using a very vanilla cause and effect realist interpretation and applying it to the case where the BSM test is done after the measurement of 1 and 4. @DrChinese says the BSM test results in 1 & 4 entanglement, but my interpretation is that the actions prior to the BSM test is what leads to the entanglement.

Here is how I describe this case from the realist perspective:
Measuring the polarization of photon 1 makes the polarizations states of 1 & 2 different. Measuring the polarization of photon 4 makes the polarization states of 3 & 4 different. When photons 2 and 3 are identical in almost all properties including polarization, the BSM test will determine they are indistinguishable and this indicates that the measurements of 1 & 4 are maximally entangled.

And here is how I describe this case differently than @DrChinese
The BSM test is what reveals the correlation, not what causes the correlation. All the actions that happen prior to the BSM test being performed is what causes the correlation between 1 & 4. If you choose not to do the BSM test, the correlation between 1 & 4 is still there, you just don't have a BSM test to select it from your measurement data.

So now that I have clarified my perspective. Is my perspective an acceptable one to hold? And if it is not acceptable, what is wrong with it?

 

[PeterDonis]

Measuring the polarization of photon 1 makes the polarizations states of 1 & 2 different. Measuring the polarization of photon 4 makes the polarization states of 3 & 4 different.

This is "realist", but it isn't local, because 1 & 2 are spatially separated when photon 1 is measured, and 3 & 4 are spatially separated when photon 4 is measured. So this is a nonlocal "action at a distance" model. You might be satisfied with that, but the others that have been objecting to the interpretation @DrChinese has been using won't be; they have been trying to defend a local interpretation, where there is no "action at a distance" even when particles are entangled, so measuring photon 1 can't change anything about photon 2, and measuring photon 4 can't change anything about photon 3.

 

[DrChinese]

I am using a very vanilla cause and effect realist interpretation and applying it to the case where the BSM test is done after the measurement of 1 and 4. @DrChinese says the BSM test results in 1 & 4 entanglement, but my interpretation is that the actions prior to the BSM test is what leads to the entanglement.

Here is how I describe this case from the realist perspective:
Measuring the polarization of photon 1 makes the polarizations states of 1 & 2 different. Measuring the polarization of photon 4 makes the polarization states of 3 & 4 different. When photons 2 and 3 are identical in almost all properties including polarization, the BSM test will determine they are indistinguishable and this indicates that the measurements of 1 & 4 are maximally entangled.

And here is how I describe this case differently than @DrChinese
The BSM test is what reveals the correlation, not what causes the correlation. All the actions that happen prior to the BSM test being performed is what causes the correlation between 1 & 4. If you choose not to do the BSM test, the correlation between 1 & 4 is still there, you just don't have a BSM test to select it from your measurement data.

If you assume Causality (as @vanhees71 does, for example), then naturally your explanation will center on what happens first. It does lead to awkward explanations (as the before/after scenarios will not be consistent).

Also, as @PeterDonis points out in post #117, your explanation is nonlocal - which is fine. QM is "quantum nonlocal." However, the usual point is to maintain Einsteinian causality in which no effect can occur outside of a light cone. So there is that.

 

[kurt101]

It can't reveal any correlation between 1 & 4 that existed before, because there is no correlation between 1 & 4 that existed before. That's obvious from the initial state that was prepared.

The initial state that 1 & 2 were prepared in and 3 & 4 were prepared in are changed prior to when the BSM test is performed. So it is not obvious to me that 1 & 4 were not changed to a maximally entangled state prior to the BSM test.

This is "realist", but it isn't local, because 1 & 2 are spatially separated when photon 1 is measured, and 3 & 4 are spatially separated when photon 4 is measured. So this is a nonlocal "action at a distance" model. You might be satisfied with that,

Yes, I am satisfied with this nonlocal "action" at a distance model.

but the others that have been objecting to the interpretation @DrChinese has been using won't be; they have been trying to defend a local interpretation, where there is no "action at a distance" even when particles are entangled, so measuring photon 1 can't change anything about photon 2, and measuring photon 4 can't change anything about photon 3.

So you are saying @DrChinese and I are not really in any disagreement? I mean we probably have some disagreements in our preferred interpretation (assuming he has one), but that set aside, are you saying he is not saying my interpretation and perspective are not allowed?

 

[PeterDonis]

The initial state that 1 & 2 were prepared in and 3 & 4 were prepared in are changed prior to when the BSM test is performed.

I deleted the post you are quoting here.

I am satisfied with this nonlocal "action" at a distance model.

Ok.

So you are saying @DrChinese and I are not really in any disagreement?

Not if you're OK with nonlocal action at a distance. See his post #118.

 

[kurt101]

If you assume Causality (as @vanhees71 does, for example), then naturally your explanation will center on what happens first. It does lead to awkward explanations (as the before/after scenarios will not be consistent).

When you refer to awkward explanations you are referring to how to describe relativistically versus absolutely?

 

[PeterDonis]

When you refer to awkward explanations you are referring to how to describe relativistically versus absolutely?

He's referring to the fact that the actual math is the same regardless of the time ordering of the measurements--as has already been repeatedly pointed out. That means that any purported "explanation" in words that changes depending on the time ordering of the measurements has a problem in explaining how it can be consistent with the math, which doesn't depend on the time ordering of the measurements.

 

[DrChinese]

When you refer to awkward explanations you are referring to how to describe relativistically versus absolutely?

What @PeterDonis said, which as usual he states better and more succinctly than I can.

 

[kurt101]

He's referring to the fact that the actual math is the same regardless of the time ordering of the measurements--as has already been repeatedly pointed out. That means that any purported "explanation" in words that changes depending on the time ordering of the measurements has a problem in explaining how it can be consistent with the math, which doesn't depend on the time ordering of the measurements.

Ok, I don't really think that is a blocker to my perspective, but I can at least understand the push back on my perspective.

My argument to support how can it be consistent is this:

The entanglement swapping experiment where BSM test is done last, run in reverse is essentially an EPR experiment. Photons 2 & 3 start out in the same state. The reverse non-local action happens between 1 & 2. The reverse non-local action happens between 3 & 4. And the 4 ends up in a state that is entangled with 1. So intuitively this explanation should not be surprising at all.

 

[vanhees71]

Nobody disagrees with the math, but I think there is considerable disagreement on how to interpret it.But how does this work? The math doesn't say. If you are satisfied with just pointing at the math and leaving any question the math doesn't answer unanswered, that's fine. But many people aren't; that's why this thread exists. And it doesn't seem like that's the position you're taking anyway: you're asserting what I quoted above as an explanation; you're not saying that no explanation is required at all.

On its face, it seems like you are describing a local hidden variable model of the kind that is ruled out by Bell's Theorem.

But what else do you need as "explanation"? Of course, the quantum formalism is pretty abstract, but we don't have anything more intuitive to express this quantum behavior. Nature doesn't care about what we find intuitive!

There is no hidden variable in what I've written above. It's just QFT, and indeed there's not the slightest hint to any hidden variables from any experiment. The important point is that QFT is "local" due to the microcausality constraint, but is not "realistic", i.e., observables don't need to take determined values

 

[PeterDonis]

The reverse non-local action happens between 1 & 2. The reverse non-local action happens between 3 & 4.

But 1 & 4 don't even exist when the BSM is done in this scenario. In the "forward" version they were measured and destroyed before the BSM was done. In the "reverse" version they would have to be created somehow (and I don't know how you would do that).

 

[vanhees71]

1. Agreed by me.

2. What does this even mean? What correlations? You agree that (14) are NOT entangled in the initial state (good, we agree). And all of them (in the ideal case) are entangled in one of the 4 Bell states in the final state (even if that particular state is unknown).

I have said this before: Certainly there is nothing that connects the successful swap to any change in the statistical relationship of the (14) pairs in *your* reading of the experiment. But we are talking about perfect correlations and violation of Bell inequalities here. According to your reading, EVERY entangled pair anywhere in the universe at any time could be shown to have the same Bell correlations with (12) or (34) - and each other. Because in your mind, those correlations are present (pre-existing) in any 2 entangled pairs - they are just waiting to be uncovered. (Since of course, nothing actual happens/changes with a BSM - in your worldview).

I guess the problem is that it's hard to say what I want to say, and English is not my mother tong. The point is that (12) is max. entangled as well as (34). Now you take 2 from one of these entangled pairs and 3 from the other and entangle them by project them to a Bell state, which necessarily also implies entanglement of (14). The entanglement swapping is possible, because of the entanglement (i.e. the stronger-than-classical correlations described by them) of (12) and of (34). In the so selected subensemble neither (12) nor (34) are entangled (that's why it's called "swapping").

How do you make any physical sense of that? Any 2 entangled pairs anywhere have these hidden correlations, waiting to be revealed? Are you really saying that?

In a sense yes. Of course for each single system it's random, whether I'm succesful or not in swapping. It's as in the somewhat simpler case of teleportation.

 

[vanhees71]

This is "realist", but it isn't local, because 1 & 2 are spatially separated when photon 1 is measured, and 3 & 4 are spatially separated when photon 4 is measured. So this is a nonlocal "action at a distance" model. You might be satisfied with that, but the others that have been objecting to the interpretation @DrChinese has been using won't be; they have been trying to defend a local interpretation, where there is no "action at a distance" even when particles are entangled, so measuring photon 1 can't change anything about photon 2, and measuring photon 4 can't change anything about photon 3.

The projection of pair (23) to a Bell state is a local measurement, i.e., the involved photons interact locally with the PBS.

 

[PeterDonis]

The projection of pair (23) to a Bell state is a local measurement, i.e., the involved photons interact locally with the PBS.

Yes. But @kurt101 is saying that this measurement also changes photons 1 and 4. That is not local. (I'm not saying you claim this, only that @kurt101 does.)

 

[DrChinese]

1. I guess the problem is that it's hard to say what I want to say, and English is not my mother tong.

2. The point is that (12) is max. entangled as well as (34). Now you take 2 from one of these entangled pairs and 3 from the other and entangle them by project them to a Bell state, which necessarily also implies entanglement of (14). The entanglement swapping is possible, because of the entanglement (i.e. the stronger-than-classical correlations described by them) of (12) and of (34). In the so selected subensemble neither (12) nor (34) are entangled (that's why it's called "swapping").

3. In a sense yes. Of course for each single system it's random, whether I'm successful or not in swapping. It's as in the somewhat simpler case of teleportation.

1. I have always thought your English is excellent, and in fact superior to most native English speakers. I have always assumed your mother tongue is German, but I guess I should ask.

2. It not that it "implies entanglement of (14)"... it is the cause of it.

3. This is the point I keep making: in the ideal case they are ALL successful swaps. If there are matched 1 & 4 clicks, a swap occurs (assuming the BSM setup is operating). You don't know the resulting state, but they are entangled. So there really is no subensemble in the sense you imagine. Once you realize that ALL of the matched 1 & 4 pairs are entangled and that there are no rejects, you realize that the "out" you are relying upon doesn't work. You could send the (23) pair into outer space (after the BS where they are made indistinguishable, never detecting them) and 1 & 4 would still be entangled - again you wouldn't know which Bell state they are in.

All of the (14) matched pairs are... entangled. So since they didn't start that way, as we agree, something changed. There is only one candidate cause: the BSM. Without that, none of the (14) matched pairs are entangled. None.

 

[kurt101]

But 1 & 4 don't even exist when the BSM is done in this scenario. In the "forward" version they were measured and destroyed before the BSM was done. In the "reverse" version they would have to be created somehow (and I don't know how you would do that).

I don't really think that you can run the entanglement swapping experiment backwards, but if you could it looks like an EPR experiment. I mean the case where the BSM test is done after the measurement of 1 & 4. Anyways this is what made me think that my explanation might actually work.

Yes. But @kurt101 is saying that this measurement also changes photons 1 and 4. That is not local. (I'm not saying you claim this, only that @kurt101 does.)

And just to be clear I am only saying that the 2 & 3 measurement changes photons 1 & 4 in the case where the BSM test on 2 & 3 is done before measuring 1 & 4. I thought this was @DrChinese position as well.

In the case where the BSM test is done after 1 & 4, I am saying that the measurement of 1 & 4 changes the photons prior to the BSM test of 2 & 3.

 

[PeterDonis]

I don't really think that you can run the entanglement swapping experiment backwards

It's not even a matter of not being able to do it in practice: you can't even build a working theoretical model of it in principle.

but if you could it looks like an EPR experiment.

No, it doesn't, because in the reversed version, photons 2 & 3 would not be entangled when they "emerge" from the "BSM" (since in the forward version they are not entangled going into the BSM). This is not even considering the fact that it is impossible to create photons 1 & 4 (which do not even exist at the time 2 & 3 "emerge" from the "BSM" in the backward versions) in just the right states to match up wtih 2 & 3.

I am only saying that the 2 & 3 measurement changes photons 1 & 4 in the case where the BSM test on 2 & 3 is done before measuring 1 & 4. I thought this was @DrChinese position as well.

@DrChinese is taking the position that the 2 & 3 BSM has the same effect (it swaps entanglements) regardless of the time ordering of the 2 & 3 vs. 1 & 4 measurement. You are not taking that position; you are saying that the BSM only has this effect if it happens first.

In the case where the BSM test is done after 1 & 4, I am saying that the measurement of 1 & 4 changes the photons prior to the BSM test of 2 & 3.

Yes, and that is not the position @DrChinese is taking. See above.

 

[kurt101]

It's not even a matter of not being able to do it in practice: you can't even build a working theoretical model of it in principle.No, it doesn't, because in the reversed version, photons 2 & 3 would not be entangled when they "emerge" from the "BSM" (since in the forward version they are not entangled going into the BSM). This is not even considering the fact that it is impossible to create photons 1 & 4 (which do not even exist at the time 2 & 3 "emerge" from the "BSM" in the backward versions) in just the right states to match up wtih 2 & 3.

Yes, you are strictly correct, but I still think there is symmetry in running the experiment backwards that looks a lot like the EPR experiment which guides intuition on why my explanation works. If you run one side in reverse, photon 2 would leave the BSM. Then when photon 1 reaches the detector (also going backwards) the spooky action in reverse would happen. Then eventually 1 & 2 would meet having the same state. So the last part if is out of order for the EPR experiment and that is a flaw in my analogy. And to make the analogy work you have to think as 1 & 2 acting together as a single photon. It is far from perfect, but it is what made me think it should work.

@DrChinese is taking the position that the 2 & 3 BSM has the same effect (it swaps entanglements) regardless of the time ordering of the 2 & 3 vs. 1 & 4 measurement. You are not taking that position; you are saying that the BSM only has this effect if it happens first.Yes, and that is not the position @DrChinese is taking. See above.

Agreed.

 

[PeterDonis]

I still think there is symmetry in running the experiment backwards that looks a lot like the EPR experiment

Unless you can justify this claim with math (which you can't), it's personal speculation and is off limits here.

 

[kurt101]

Unless you can justify this claim with math (which you can't), it's personal speculation and is off limits here.

I can write a program which I tend to think of the program as being the math, but it is not the language of math that you are looking for. Correct?

And even if I was able to translate the program to math that was acceptable, I imagine it would still fall under the grounds of speculative theory.

So unless I get something published and peer reviewed, I think I have gone as far as I can here.

I feel as if I have gotten my question answered and understand @DrChinese position.

 

[PeterDonis]

I can write a program

That's not the math of QM, that's just some program you wrote. I am talking about the standard math of QM.

even if I was able to translate the program to math that was acceptable

Or you could just use the standard math of QM. That's what it's for.

 

[Morbert]

Regarding precisely "when" (14) pairs become entanglemed in the BSM experiment according to the minimalist* interpretation: The point where the minimalist interpretation makes contact with reality is in macroscopic preparations and macroscopic outcomes. Quantum states, Hermitian operators etc do not characterise the microscopic system. They instead characterise macroscopic interventions on the microscopic system. In statistical language, projectors mentioned previously in this thread like $\{\mathbb 1_1 \otimes P_{23,i} \otimes \mathbb 1_4\}$ that project to the BSM basis, select for subensembles according to BSM outcome. No other element of reality beyond the macroscopic BSM events is considered by the theory. If the experiment is modified to include a BSM on (14) as well, then projectors like $\{\mathbb 1_2 \otimes P_{14,i} \otimes \mathbb 1_3\}$ will select for subensebles according to this BSM outcome. Reality is not characterised by some time-evolution process whereby (14) entanglement is induced at some moment at or after preparation. Reality is instead *only* characterised by a collection of experimental outcomes that followed from identical preparations, with computable frequencies and correlations. Locality is readily preserved.

*By minimalist I am referring to the account of QM presented by Fuchs and Peres in articles like this one https://physicstoday.scitation.org/doi/pdf/10.1063/1.883004

 

[Morbert]

PS re/ realism and locality - For posterity I will repost the obligatory video by Gell-Mann that talks about the recovery of a local realist interpretation of measurement as revealing pre-existing properties, without recourse to hidden variables. https://www.webofstories.com/play/murray.gell-mann/165

 

[PeterDonis]

the minimalist* interpretation

This seems to be more or less the interpretation that @vanhees71 is using.

 

[Morbert]

This seems to be more or less the interpretation that @vanhees71 is using.

I think so too. The only possible difference is that Fuchs and Peres are also willing to talk about the likelihood of single events as well as frequencies of events in ensembles, which I am not sure vanhees is willing to do.

"When we are told that the probability of precipitation tomorrow is 35%, there is only one tomorrow." -- Fuchs and Peres