Bell vs Kolmogorov: Unravelling Probability Theory Limits

[martinbn]

It was referring to your demand of giving a description of the whole setup as an initial-value problem.

Ok, but I want to see it done.

 

[vanhees71]

But it's done in quantum-optics textbooks. My favorites are

J. Garrison and R. Chiao, Quantum optics, Oxford University
Press, New York (2008),
https://doi.org/10.1093/acprof:oso/9780198508861.001.0001

M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge
University Press (1997).

 

[stevendaryl]

One thing that bothers me about EPR is the use of "predict". If you can predict with 100% certainty then it has to be an element of reallity. But the prediction of Alice is not for the future of her measurment!

I don’t understand this objection. Alice and Bob will get together later and tell each other what results they got. She’s making a prediction about what Bob will say.

I guess we can coin a new term, something like “teledict”, which would be defined as computing some fact about conditions that are not in your backwards lightcone (whether or not they are inthe future.

 

[martinbn]

I don’t understand this objection. Alice and Bob will get together later and tell each other what results they got. She’s making a prediction about what Bob will say. I guess we can coin a new term, something like “teledict”, which would be defined as computing some fact about conditions that are not in your backwards lightcone (whether or not they are inthe future.

It is not an objection but something that bothers me because it is not clear to me. Changing it to what Bob would say is not really an improvement. Between Bob's measurement and the meeting with Alice there is nothing strange or unusual about the reality of the measurement outcome. So the question really is what will Bob measure.

What I don't understand is the following. If you can predict with 100% certainty the value of a dynamical variable, then the measurment only reviels a preexisting value, and the theory should account for that. That is how I understand EPR's what is reasanable to consider as an element of reality. And they argue that this applies to Bob's measurment. But for me to predict the outcome of Bob's measurement means that given all the data in the past light-cone of that event you can calculate the value Bob will get. And that is not possible. What they do is to look at Alice's outcome and inffer Bob's. But that is not a prediction. It is almost like looking at the answer before saying what the answer is. Of course it is not quite like that, so I am not objecting to anything, I am only stating what bothers me. The two possible outcome for Alice and Bob are the pairs {1,-1} and {-1,1} . So if you see the first half of the pair you can "predict" the other, so!

 

[Killtech]

In the EPR argument, the claim that measurements had pre-existing values was not an assumption, it was the conclusion of their argument. And I don’t think that the argument implicitly or otherwise assumes classical pointlike particles.

Certainly even classically, measurements don’t necessarily reveal a pre-existing value. The value of a measurement might be some kind of cooperative result of the interaction between the system being measured (the “measuree” to coin a term) and the system doing the measurement (the “measurer”). For example, if you are doing a survey to find out if someone is pro- or anti- gun control, the answer you get from them may depend on how the question is asked. I’m not sure that this is exactly what is meant by “contextuality”, so let me just call it an “emergent value ”, since the value emerges through the interaction between the “measurer” and the “measuree”.

So the question arises: is quantum uncertainty due to measurement results being emergent in this sense?

Einstein et al argued that it can’t be.

Alice and Bob are two experimenters who are far away from each other, performing measurements. Alice performs her measurement and gets a result. On the basis of her result, she knows with 100% certainty what result Bob will get (assuming he told her ahead of time which measurement he would perform).

So Einstein et al reasoned that there can’t be anything “emergent” about Bob’s result. If the result depended on details about Bob and his measurement device and exactly how the measurement was performed, then how can Alice predict with 100% certainty what result Bob will get, since she doesn’t know any of those details? She only knows (1) What measurement Bob will perform (because he told her ahead of time) and (2) what the result of her measurement was. No other details about Bob’s situation are relevant. Einstein thought that this situation could only be explained by the result being pre-determined. That was the conclusion, not the assumption.

There is nothing about particles or things being pointlike.

If you present it like that, then i am sure i follow the argument there.

The argument against an emergent behavior can only be made on the grounds of the spatial separation and nothing else - and as such it already based on an assumption only.

On the other hand, any attempt to understand the behavior of Bell experiments by studying its statistics leads to an requirement of emergent behavior: we can just write down all outcomes into a logic table and each such logic table can be implemented via a logic circuit that realizes this statistic. We will find that any such implementation is fundamentally emergent - and therefore it runs into issues when the decisions of Alice and Bobs are delayed until the last possible moment and the time in between them is faster then the travel time of logical signals within the circuit.

The statistical analysis via a logic circuit represents well the way classic Kolmogorov probability theory views Bells results. So if we want to preserve it, it requires us to postulate an emergent behavior i.e. somethin akin to a "wave function" collapse interpreted in terms of some physical object collapsing.

But we know so far that no experiment was able to provide evidence for either interpretation, hence as of now both options remain just equivalent alternative interpretations and non can be excluded.

 

[vanhees71]

What I don't understand is the following. If you can predict with 100% certainty the value of a dynamical variable, then the measurment only reviels a preexisting value, and the theory should account for that. That is how I understand EPR's what is reasanable to consider as an element of reality. And they argue that this applies to Bob's measurment. But for me to predict the outcome of Bob's measurement means that given all the data in the past light-cone of that event you can calculate the value Bob will get. And that is not possible. What they do is to look at Alice's outcome and inffer Bob's. But that is not a prediction. It is almost like looking at the answer before saying what the answer is. Of course it is not quite like that, so I am not objecting to anything, I am only stating what bothers me. The two possible outcome for Alice and Bob are the pairs {1,-1} and {-1,1} . So if you see the first half of the pair you can "predict" the other, so!

The important point is that the violation of Bell's inequality together with the assumption of locality (in the sense that no faster-than-light influence of A's measurement event at B's measurement event and vice versa, if the measurement events are space-like separated) rules out that the measurement outcomes are predetermined. To the contrary both A and B measure simply the polarization of exactly unpolarized photons (I guess the preparation of such entangled pairs provides the most accurate source of single unpolarized photons possible).

According to Q(F)T however, in the entangled photon state $|HV \rangle-|VH \rangle$, there are the 100% correlations between the outcome of A's and B's measurement (provided they measure the polarization in precisely the same or precisely perpendicular directions), i.e., they are there due to the preparation of the photon pairs in this state. So indeed if A measured H, she immediately knows that B will find V for his photon, and that's what's indeed confirmed by experiments, but of course you can only confirm this correlation by comparing the measurement protocols after both measurements were done. So there is no contradiction between locality in the sense of QFT ("microcausality") and the 100% long-ranged correlations between the two photons due to their preparation in an entangled state.

 

[Killtech]

What I don't understand is the following. If you can predict with 100% certainty the value of a dynamical variable, then the measurment only reviels a preexisting value, and the theory should account for that.

The way you formulate it sounds deeply flawed. Like take the statement and apply it to any deterministic theory which in theory makes predictions about the future with 100% certainty. This view would strictly imply the future to be already preexisting for such a theory.

But surely we can agree that is not a general requirement for deterministic theories.

 

[stevendaryl]

The way you formulate it sounds deeply flawed. Like take the statement and apply it to any deterministic theory which in theory makes predictions about the future with 100% certainty. This view would strictly imply the future to be already preexisting for such a theory.

Not the future, but the value. If you can predict with 100% certainty the location of a particle 10 years from now, then that means that “the location 10 years from now” is a function of the current state.

 

[stevendaryl]

The two possible outcome for Alice and Bob are the pairs {1,-1} and {-1,1} . So if you see the first half of the pair you can "predict" the other, so!

Yes, but I don’t understand what point you are making.

Let me introduce a fictional situation and see what you think about it. Suppose that there are a pair of coins. Each coin can be flipped to give a result of “heads” or “tails”. Looking at either coin in isolation reveals no pattern to the results, other than 50/50 chance for each outcome. Yet comparing the two coins shows that the nth flip of one coin always gives the opposite result of the nth flip of the other coin.

I think that most people confronted with such a coin would assume that either the results are predetermined, or that there is some kind of long range interaction between the coins.

 

[Killtech]

Not the future, but the value. If you can predict with 100% certainty the location of a particle 10 years from now, then that means that “the location 10 years from now” is a function of the current state.

where the "current state" is unusually a collection of values relating to various locations, i.e. non local data.

So in the thinking EBR such a "current state" would only be allowed to contain data from within the specific light cone? I suppose that EBR had something like a particle picture in mind where they though of something like local hidden variables for each particle that had the required information. Their whole argumentation makes a lot more sense with this in mind. But as of today we know that no such idea will work.

On the other hand, if the "current state" isn't restricted to light cone then there is no issue to view Bells result as emergent.

 

[DrChinese]

It is not an objection but something that bothers me because it is not clear to me. Changing it to what Bob would say is not really an improvement. Between Bob's measurement and the meeting with Alice there is nothing strange or unusual about the reality of the measurement outcome. So the question really is what will Bob measure.

What I don't understand is the following. If you can predict with 100% certainty the value of a dynamical variable, then the measurment only reviels a preexisting value, and the theory should account for that. That is how I understand EPR's what is reasanable to consider as an element of reality. And they argue that this applies to Bob's measurment. But for me to predict the outcome of Bob's measurement means that given all the data in the past light-cone of that event you can calculate the value Bob will get. And that is not possible. What they do is to look at Alice's outcome and inffer Bob's. But that is not a prediction. It is almost like looking at the answer before saying what the answer is. Of course it is not quite like that, so I am not objecting to anything, I am only stating what bothers me. The two possible outcome for Alice and Bob are the pairs {1,-1} and {-1,1} . So if you see the first half of the pair you can "predict" the other, so!

If someone (Alice) hands the correctly predicted answer to what Bob will observe, that is an EPR "element of reality" - and you already acknowledge that. They *assume* (explicitly) that whoever/however (Alice) came up with that answer, did NOT affect what Bob will observe (of course that is subject to challenge). Thus the answer MUST have been predetermined.

So it seems to me your (non-)objection - that QM does not supply a way to predict that answer from the initial conditions - is not relevant to their particular argument. After all, perhaps those mechanics will perpetually be obscured to us. The important item is whether you agree that Bob's outcome was predetermined (using their argument*).*Which also has the explicit assumptions of a) locality and b) realism (being the simultaneous existence of all identifiable "elements of reality"). This is of course what Bell hopped on.

 

[martinbn]

Yes, but I don’t understand what point you are making.

Let me introduce a fictional situation and see what you think about it. Suppose that there are a pair of coins. Each coin can be flipped to give a result of “heads” or “tails”. Looking at either coin in isolation reveals no pattern to the results, other than 50/50 chance for each outcome. Yet comparing the two coins shows that the nth flip of one coin always gives the opposite result of the nth flip of the other coin.

I think that most people confronted with such a coin would assume that either the results are predetermined, or that there is some kind of long range interaction between the coins.

Yes, that is what most/all will assume that. So?

 

[martinbn]

If someone (Alice) hands the correctly predicted answer to what Bob will observe, that is an EPR "element of reality" - and you already acknowledge that. They *assume* (explicitly) that whoever/however (Alice) came up with that answer, did NOT affect what Bob will observe (of course that is subject to challenge). Thus the answer MUST have been predetermined.

So it seems to me your (non-)objection - that QM does not supply a way to predict that answer from the initial conditions - is not relevant to their particular argument. After all, perhaps those mechanics will perpetually be obscured to us. The important item is whether you agree that Bob's outcome was predetermined (using their argument*).*Which also has the explicit assumptions of a) locality and b) realism (being the simultaneous existence of all identifiable "elements of reality"). This is of course what Bell hopped on.

No, not relevant to their argument. My non-objection is about their definition of elements of reality.

 

[stevendaryl]

Yes, that is what most/all will assume that. So?

So that’s the intuition behind Einstein’s argument.

What Einstein seemed to believe is that what future results of a measurement are possible is determined by the state of the universe prior to the measurement. Furthermore, if all causal influences are limited by lightspeed, then the possibilities are in fact determined by just facts about the universe that are available in the backwards light cone.

The EPR experiment seems to violate this intuition, because Alice getting spin-up (in the anti-correlated half-spin case) means that spin-up is not a possible result for Bob, even though no facts available in his backwards light cone could tell him this.

 

[stevendaryl]

So in the thinking EBR such a "current state" would only be allowed to contain data from within the specific light cone? I suppose that EBR had something like a particle picture in mind where they though of something like local hidden variables for each particle that had the required information.

I don’t see what it has to do with particles. The question is: Does Alice’s measurement affect the “current state”, or not? If so, then her actions affect Bob’s results, contrary to the limitations of relativity.

 

[Killtech]

I don’t see what it has to do with particles. The question is: Does Alice’s measurement affect the “current state”, or not? If so, then her actions affect Bob’s results, contrary to the limitations of relativity.

So Alice measurements affect the "current state". This isn't contrary to relativity, since this change of current state isn't locally measurable. Relativity makes no statements for such things. It restricts anything that can be used as a signal which are inherently locally accessible.

 

[martinbn]

So that’s the intuition behind Einstein’s argument.

What Einstein seemed to believe is that what future results of a measurement are possible is determined by the state of the universe prior to the measurement. Furthermore, if all causal influences are limited by lightspeed, then the possibilities are in fact determined by just facts about the universe that are available in the backwards light cone.

This is exactly what confuses me. Future and prior in the relativistic setting are not the same as in the nonrelativistic. The measurement of Alice is not prior to that that of Bob's, nor is his in the future of that of Alice.

The EPR experiment seems to violate this intuition, because Alice getting spin-up (in the anti-correlated half-spin case) means that spin-up is not a possible result for Bob, even though no facts available in his backwards light cone could tell him this.

But this is not strange. Look at this: you flip a coin, and I take a look at it. If it is heads it is impossible for you to see tails and nothing in the past of the flip can tell you that. Of course it is not exactly the same, but it looks similar. Almost as if knowing the answer allows me to predict it with 100% certainty.

 

[PeterDonis]

So Alice measurements affect the "current state". This isn't contrary to relativity, since this change of current state isn't locally measurable. Relativity makes no statements for such things.

No, you are misstating this. What relativity says is that there can be no such thing as a "current state" that includes spacelike separated events. So the view of QM you are using is fundamentally incompatible with relativity, since it makes use of a "current state" that does include spacelike separated events. Since the view of QM you are using is based on non-relativistic QM, that is not surprising; but it doesn't mean you can just wave your hands and use it in a context where relativity is clearly relevant.

You need to use relativistic QM in this context, and the only relativistic QM we have is quantum field theory. Quantum field theory says that spacelike separated measurements must commute, i.e., their results cannot depend on the order in which they are made. And that means that any claims about Alice's measurement affecting Bob's or Bob's affecting Alice's cannot be right in the context of QFT, since the measurement results are independent of their order so neither one can possibly affect the other.

 

[Killtech]

No, you are misstating this. What relativity says is that there can be no such thing as a "current state" that includes spacelike separated events.

But no such thing is required. If you think of events, the "current state" partly contains the event of calculating the correlation between Alice and Bob - but partly. The current state changes in what we can say about the possible correlations but since Alice "current state" does not yet include the information for what measurement Bob will ultimately settle for - because that event is indeed spacelike separated. Partial information makes it not a full event until it is combined with supplementing partial information and only then it can be related to other events. But the combination to a full event cannot happen outside either Alice nor Bobs light cone so it never contains any spacelike separated information.

So you can interpret it as Alice knows how she impacted Bobs result, but without knowing what Bob's input will be, she cannot calculate the outcome. It also means that the emergent behavior becomes relative depending on the observer, which is fine for as long as no FTL signaling can happen.

 

[PeterDonis]

no such thing is required

Then you should not be using the term "current state" at all, since that term does require that spacelike separated events are part of the "current state".

the "current state" partly contains the event of calculating the correlation between Alice and Bob - but partly

This is just another way of saying that the "current state" contains spacelike separated events. Even if it only "partly" does so, that's enough to contradict relativity. It's also not anywhere in QFT. If you disagree, please show me where in QFT this "current state" you speak of is.

So you can interpret it as Alice knows how she impacted Bobs result

In QFT, Alice cannot "impact" Bob's result. Alice's and Bob's measurements are spacelike separated, which means they must commute, which means neither one can "impact" the other's result.

Again, if you disagree, please show me where in QFT this "impact" you speak of is. You can't just wave your hands here. We're talking about physics, and physics makes predictions with math, not vague handwaving. So you need to back up your claims with math.

It also means that the emergent behavior becomes relative depending on the observer

What does this even mean? Alice's and Bob's measurement settings and measurement results are perfectly objective; they are the same for all observers (though not all observers gain information about them at the same time).

 

[Killtech]

Then you should not be using the term "current state" at all, since that term does require that spacelike separated events are part of the "current state".

i used the "" for a reason and its a word @stevendaryl used which i originally replied to.

This is just another way of saying that the "current state" contains spacelike separated events. Even if it only "partly" does so, that's enough to contradict relativity. It's also not anywhere in QFT. If you disagree, please show me where in QFT this "current state" you speak of is.

Then you clearly misunderstand. The difference is that before the measurement Alice would have described the system to be in $|HV\rangle+|VH\rangle$ whereas afterwards she would says its $|H\rangle|V\rangle$ assuming she got an $H$ result. This information is not spacelike separated since the $|V\rangle$ she has for Bobs photon isn't a measured value at Bob's location but represents her view of the photon.

What does this even mean? Alice's and Bob's measurement settings and measurement results are perfectly objective; they are the same for all observers (though not all observers gain information about them at the same time).

It means that the change of state Alice registers is not purely an update of knowledge about the unknows of the system and does not transform as such information would otherwise do. It instead additionally includes some change of the system, rendering some previous knows unknow, specifically she impacted/changed the axis at which Bobs results will be deterministic/maximally anticorrelated.

 

[vanhees71]

Another point of view is much more natural, and it's best "visualized" using the Heisenberg picture of time evolution: The state describes the preparation procedure, i.e., in this case the two-photon polarization state as the Bell state $\hat{\rho}=|\Psi \rangle \langle \Psi|$, whose state ket is
$$|\Psi \rangle=\frac{1}{\sqrt{2}} (|HV \rangle-|VH \rangle).$$
In the Heisenberg picture this state and the state ket are time-independent. This implies that, if A measures the polarization of her photon to be $H$ then she immediately knows that B will find $V$, provided he also measures the polarization in the same direction as she did with her photon, i.e., the 100% correlation for this particular case of joint measurements of the polarization of both photons is already imprinted by the state preparation procedure, leading to the said Bell state for the two photons. This view is confirmed in experiments, where the registration events of the photons and thus the measurements of their polarization are space-like separated, such that there cannot be any influence of one measurement on the other (according to local, microcausal QFT which describes photons very well since QED is the most accurate theory ever), i.e., although the single-photon polarizations are utmost indetermined (they are exactly unpolarized photons) before A's or B's measurement, the 100% correlation is still implied by the description of the entangled Bell quantum state.

 

[PeterDonis]

i used the "" for a reason and its a word @stevendaryl used which i originally replied to.

Unlike you, @stevendaryl correctly stated that if Alice can affect the "current state", that is contrary to relativity. You claimed it isn't. Your claim is wrong. My statements about the term "current state" were part of explaining why your claim is wrong. @stevendaryl didn't make your claim, you did.

Then you clearly misunderstand.

No, you misunderstand. The whole notion of a "state" that includes both Alice's and Bob's qubit, since they are spatially separated, is not consistent with relativity. If you are going to try to describe what happens in a way that is consistent with relativity, you cannot use these kinds of states at all. You need to use a correct QFT description of what is going on.

It means that the change of state Alice registers is not purely an update of knowledge about the unknows of the system and does not transform as such information would otherwise do. It instead additionally includes some change of the system, rendering some previous knows unknow, specifically she impacted/changed the axis at which Bobs results will be deterministic/maximally anticorrelated.

I don't know where you're getting all this from, but it's not from any correct understanding of QM and relativity. It's not even consistent with what you said earlier in the same post.

 

[PeterDonis]

The state describes the preparation procedure

Which took place at a single, localized spacetime event, so it does not raise the difficulties that a "state" that purports to describe both qubits after they are spatially separated would.

 

[vanhees71]

Exactly! In this case it's not so clear what "spatially separated" really means since you cannot even localize photons theoretically, because they do not have a position observable to begin with. All you can say is that a detector at some place registered a photon at some time (that's what I mean when I write "detection event"). Together with some optical elements you can measure its polarization (e.g., using a polaroid filter or a polarizing beam splitter).

 

[PeterDonis]

In this case it's not so clear what "spatially separated" really means

Alice's and Bob's measurement events are spatially separated, so the two qubits they measured were spatially separated, at the very least, when they were measured. I agree that localizing photons is problematic, but photons are not the only possible realizations of qubits; we could use electron spins, for example, where the localization issues don't arise and we can unproblematically talk about the electrons being spatially separated after they are prepared and while they are en route to being measured by Alice and Bob.

 

[vanhees71]

Yes, but that doesn't even matter, because the detection events are local and since they are spatially separated one cannot influence causally the other, and that's true also within relativistic local QFT (in this case QED) due to the microcausality property used in its very conceptual foundation.

 

[Killtech]

Alice's and Bob's measurement events are spatially separated, so the two qubits they measured were spatially separated, at the very least, when they were measured. I agree that localizing photons is problematic, but photons are not the only possible realizations of qubits; we could use electron spins, for example, where the localization issues don't arise and we can unproblematically talk about the electrons being spatially separated after they are prepared and while they are en route to being measured by Alice and Bob.

However, the measurement of the correlation between Bob and Alice is another matter. This event differs drastically from other events as it does not form a spacetime point but rather a hypersurface where it can obtained initially. This is because it cannot be measured at a single spacetime point i.e. locally but requires two separate measurements to obtain it. The operator that represents such an observable is therefore strictly non-separable, i.e. non local. The correlation of two spatially separated entities itself isn't a local quantity yet it is an "element of reality".

 

[PeterDonis]

the measurement of the correlation between Bob and Alice is another matter. This event differs drastically from other events as it does not form a spacetime point but rather a hypersurface where it can obtained initially.

No, it doesn't. The measurement of the correlation requires the information about the two measurement results to be brought to the same spacetime event and compared there.

This is because it cannot be measured at a single spacetime point i.e. locally but requires two separate measurements to obtain it.

Wrong. The measurement of the correlation is a single measurement that compares results from Alice and Bob. It is not the same as Alice's or Bob's measurements themselves.

The operator that represents such an observable is therefore strictly non-separable, i.e. non local.

If you think there is such a "non-separable", "non-local" operator for "measurement of the correlation", please show your math and give a reference to back it up.

 

[Killtech]

No, it doesn't. The measurement of the correlation requires the information about the two measurement results to be brought to the same spacetime event and compared there.

And there you have it: the measurement has a new "bringing" part with isn't part of any local measurement. You forget that if no one tells Alice and Bob to signal their results to a specified location... then the the correlation becomes unmeasurable everywhere. Hence, the signaling part is non-omittable and needs to be factored in. The correlation is therefore technically a function to everything the signals encountered on their way to the point where they were combined.

If you think there is such a "non-separable", "non-local" operator for "measurement of the correlation", please show your math and give a reference to back it up.

It can be directly derived from expectation value of the correlation: it's the sum of the probability of each possible state times the value it yields for the correlation. So its operator is the sum of the projection operators onto those states time the correlations they will yield. Since the possible states are continous, this is an integral.