I How "spooky action...." may work?

[bhobba]

Indeed there is nothing mysterious in how to calculate the correlations for an entangled pair, QM gives a clear recipe. Nonetheless, I find that nature is in accord with those correlations extremely mysterious and I am in good company. Don't you find that masses experience gravitational attraction mysterious in spite of knowing how to calculate the force.

I think before making statements like that a bit of thought needs to be put into the nature of explanation. An explanation assumes some things to explain others. Every explanation, every single one, has that 'mysterious' aspect to it. Its how you react to it that determines your attitude - its very personal and not science.

Regarding gravity - GR explains that attraction as the result of space-time curvature which in modern times is known to be more or less implied by the very intuitive principle of no prior geometry - why should nature single out one geometry over another? Still its an assumption and how you react to it determines if its mysterious or not - personally for me its not mysterious - but that's just me - although I suspect the vast majority would feel that way as well.

I post this a lot because I think its very important (those that have seen before just ignore it - its purely to make a point):
https://arxiv.org/pdf/quant-ph/0101012.pdf

QM can be presented in such a way, like the principle of no prior geometry, so its not 'mysterious'. From that the different kinds of statistical correlations follow. In particular as the above paper shows its the requirement of continuous transformations between pure states that takes the place of no prior geometry. It turns out that is equivalent to having entanglement:
https://arxiv.org/abs/0911.0695

Its entirely how you view and react to it - 'mysterious' is a human reaction - nature doesn't care a hoot and certainly science doesn't.

Thanks
Bill

 

[Ilja]

But why violate Einstein?

Because else you can prove Bell's inequality. All we need for this is the EPR criterion of reality: "If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity. " And Einstein causality to show that the measurement by Alice does not in any way disturb Bobs part of the system.

 

[bhobba]

But why violate Einstein? Before those two measurements, there is the quantum superposition of potential spacetime worlds - in each world, all the observed/observable events are in the most perfect agreement with Einstein causality. The two measurements together make the choice of the one actual world, one measurement just reducing the choice for another.

Virtually all of that is interpretive supposition. The QM formalisn says nothing about potential space time worlds etc etc.

Take on board the writings of the early pioneers with caution - things have moved on a lot since then.

Before forming any views on QM study a good modern text like Ballentine.

Thanks
Bill

 

[Zafa Pi]

I think before making statements like that a bit of thought needs to be put into the nature of explanation. An explanation assumes some things to explain others. Every explanation, every single one, has that 'mysterious' aspect to it. Its how you react to it that determines your attitude - its very personal and not science.

Regarding gravity - GR explains that attraction as the result of space-time curvature which in modern times is known to be more or less implied by the very intuitive principle of no prior geometry - why should nature single out one geometry over another? Still its an assumption and how you react to it determines if its mysterious or not - personally for me its not mysterious - but that's just me - although I suspect the vast majority would feel that way as well.

I post this a lot because I think its very important (those that have seen before just ignore it - its purely to make a point):
https://arxiv.org/pdf/quant-ph/0101012.pdf

QM can be presented in such a way, like the principle of no prior geometry, so its not 'mysterious'. From that the different kinds of statistical correlations follow. In particular as the above paper shows its the requirement of continuous transformations between pure states that takes the place of no prior geometry. It turns out that is equivalent to having entanglement:
https://arxiv.org/abs/0911.0695

Its entirely how you view and react to it - 'mysterious' is a human reaction - nature doesn't care a hoot and certainly science doesn't.

Thanks
Bill

How does mass pull off the warping of space?
The Hardy "axioms" to mathematicians are incomprehensible fluff, at least all the ones I've showed them to.
As far as we know nature doesn't give a hoot, nor does a specific theory/ model. But science as a human endeavor does care.
Is there any scientific result you find mysterious?

 

[bhobba]

How does mass pull off the warping of space?

If you examine a simple model of dust and assume a pseudo-riemannian geometry (that's the no prior geometry idea) then the equations of GR pretty much follow. Most books on GR explains it, at least the ones I have read, but here is not the place to discuss it - the relativity subsection is.

The Hardy "axioms" to mathematicians are incomprehensible fluff, at least all the ones I've showed them to.

Well my background is math as well and that's not my view. But obviously its a matter of opinion.

As far as we know nature doesn't give a hoot, nor does a specific theory/ model. But science as a human endeavor does care. Is there any scientific result you find mysterious?

Of course eg the Feynman sum over history approach explains the principle of least action in classical physics, but why can QFT also be put into that form. That's just one example of course.

Thanks
Bill

 

[Markus Hanke]

A statistical correlation which one cannot explain with a common cause in the past.

I don't really understand this statement - the "common cause in the past" would be the initial interaction between the two particles which created the entanglement in the first place. After that initial interaction, no further "remote action" is needed or implied to uphold the relationship. The Bell inequalities are violated in this context precisely because no real, local "hidden variables" are needed to explain entanglement. This is not just conjecture, but pretty much an empirical finding.

Maybe it's just me, but after some initial study of quantum theory, there is nothing about entanglement that seems in any way unexplainable or mysterious, unless of course one insists that the world must be both real and at the same time Einstein-local, in a classical sense. But we already know ( from empirical finding ) that this is not the case.

 

[Ilja]

I refer here to Reichenbach's common cause principle. Roughly, if we observe a correlation, it should have a causal explanation. This causal explanation may be a direct one, A->B or B->A, but it may be also explained by a common cause C, with C->A and C->B.

This common cause principle is the base for, say, conclusions about the idea that smoking causes cancer. Without Reichenbach's common cause principle, the tobacco industry could simply say "oh, this is only a correlation, this does not prove any causal relation, so what". With the common cause principle, they are forced to find some alternative common cause explanations, else, the straightforward smoking->cancer explanation would have to be accepted. They are unable to present such common cause explanations - that means, the smoking->cancer explanation is what we believe.

Fortunately, if something is a common cause explanation or not has a simple precise meaning in probabilities: P(AB|C) = P(A|C)P(B|C). So, we do not have to accept vague feelings that blablabla is sufficient as an explanation.

And now we have the 100% correlation if we measure at A and B the same direction. Once you want to explain it with a common cause at the time of the creation of the pair, you have no other choice as to postulate that this common cause C defines, predefines, the result of the measurement. And, once this common cause cannot know what direction is measured, it has to predefine them all. And, so, we obtain the functions A(a,l), B(b,l) which describe how this common cause l predefines the measurement results. And then you can prove Bell's theorem, which is empirically falsified.

 

[Markus Hanke]

I am not an expert on this matter, so someone please correct me if I am wrong, but it would seem to me that the common cause principle is not fundamental to the universe, in the sense that it is not implied by any law of nature that I am aware of. In fact, I can immediately think of a number of circumstances where it is explicitly violated - for example, if you have exactly two electrons on the same atomic shell, their spins must be opposite as per the Pauli exclusion principle. This is a statistical correlation, since, if you perform a measurement on one of them, the outcome ( spin-up or spin-down ) is probabilistic, and not predetermined; however, once the outcome has been obtained on one electron, it is necessarily determined for the other electron as well, so as to stay true to Pauli's principle. There is no causal explanation for this, only the correlation implied by the laws of quantum theory.

Please don't misunderstand me here - I don't attempt to declare the common cause principle null and void, as it is quite useful and valid under certain circumstances, as you have demonstrated with the smoker example. I merely urge caution, in that I don't think this principle is universally applicable. Specifically, I don't think it applies to quantum entanglement, nor do I think it is needed in that context.

 

[Ilja]

I do not claim it is a law of nature, I think it is a base for doing scientific research, a necessary part of the scientific method. If causal explanations for observable correlations do not exist, science is as useless as collecting stamps.

If it is not universally applicable, it means that science is not universally applicable as a method to study the world around us. So, we can do science if we study human behavior or so, but to do science in fundamental physics would be a meaningless loss of time, all we can do there is to observe correlations and to do astrology based on such observed correlations without explanation.

In some sense, there are interesting parts of human thought which are not open to the scientific method already today. Like ideas about an immortal soul and so on. Even whole philosophical concepts like fatalism. So we cannot exclude, out of principle, that the quantum domain is also inaccessible to science, and accessible only to the observation of unexplainable correlations and subsequent astrology. But at least up to now I see no evidence for this. There exist realistic and causal interpretations of quantum theory. The only reason to reject them is that they violate a nice and reasonable but nonetheless metaphysical idea that relativistic symmetry is fundamental, instead of being only a large distance approximation valid only for some quantum equilibrium states.

 

[Markus Hanke]

If causal explanations for observable correlations do not exist, science is as useless as collecting stamps.

I agree with you, but I am not sure how this contradicts how quantum entanglement is commonly understood. There is a causal explanation for entanglement, but that "cause" is not an instantaneous physical action across arbitrarily large spatial distances, but rather an interaction between the particles in the past. In other words, the particles' histories are not independent of one another, so neither are the outcomes of measurements performed on them. This doesn't necessarily imply that those measurements are deterministic ( they aren't ), nor does it necessarily imply any type of interaction between the particles at the time the measurement takes place. At the same time, there is still a common cause for the correlation between the measurement outcomes, and that is their initial interaction in the past. I do not really see a contradiction here.

 

[rubi]

There is good reason to reject Reichenbach's principle in quantum theory and it has nothing to do with locality. Reichenbach's principle is based on ordinary probability theory, which needs a simplicial state space for its formulation. However, we know that the laws of physics are governed by a theory with a non-simplicial state space (quantum theory), so it would be unreasonable to apply concepts that only make sense in the context of ordinary probability theory to it.

The unapplicability of Reichenbach's principle in the context of quantum theory doesn't preclude the existence of a common cause. It only means that a criterion that captures our classical intuition about common causes isn't adequate to also capture the notion of a common cause in the generalized framework of quantum theory. It means that we don't have probabilistic method to identity what qualifies as a common cause. Unfortunately, nature just didn't provide us with such a method. Nevertheless, a common cause can still exist.

 

[Ilja]

... but I am not sure how this contradicts how quantum entanglement is commonly understood. There is a causal explanation for entanglement, but that "cause" is not an instantaneous physical action across arbitrarily large spatial distances, but rather an interaction between the particles in the past.

If this would be a valid causal explanation, in agreement with Reichenbach's common cause and Einstein causality, we could prove Bell's theorem. Which is falsified by observation. So that your causal explanation is falsified by observation.

This doesn't necessarily imply that those measurements are deterministic ( they aren't ), nor does it necessarily imply any type of interaction between the particles at the time the measurement takes place. At the same time, there is still a common cause for the correlation between the measurement outcomes, and that is their initial interaction in the past. I do not really see a contradiction here.

That's not the point. Bell's formula
P(a,b) = \int d\lambda \rho(\lambda) A(a,\lambda) B(B,\lambda)
is also only a statistical formula, which includes a probability distribution, and has excluded any type of interaction between the particles at the time of measurement.

You have to understand that this formula follows from assuming a common cause by the interaction between the particles in the past.

Let's try. Common cause explanation by C for a 100% correlation between X, Y means P(X and Y|C) = P(X|C) P(Y|C).
X: measurement at A of a gives 1. Y: measurement at B of a gives 1. Once X and Y would violate the conservation law, we have P(X and Y) = 0, thus, also P(X and Y|C)=0. Thus, or P(X|C) = 0 or P(Y|C) = 0. If P(Y|C) = 0, it means the result at B is predefined by C as -1, which we can name simply "not Y". This gives, for 0 = P(not X and not Y|C) = P(not X|C), so, X is predefined by C too. And this works for all directions a. The free will of the experimenters to choose a and b, which decision is not known at preparation time, proves that the common cause C, even if hidden, can depend only on a probability distribution which does not depend on a and b. This is all we need for the formula above. Which is sufficient to derive a false conclusion, Bell's inequality.

 

[Ilja]

There is good reason to reject Reichenbach's principle in quantum theory and it has nothing to do with locality. Reichenbach's principle is based on ordinary probability theory, which needs a simplicial state space for its formulation.

No. Ordinary probability theory needs nothing but elementary logic. Read Jaynes, Probability theory: The logic of science.

You can always transform the space of some probability theory into a simplicial one. It is a trivial exercise. Simply postulate that the wave functions (the "pure states" of the non-simplicial theory) are the "hidden variables". That means that you define your "hidden variable" theory has the simplex created by the original pure states.

Above theories are equivalent, because in above theories all states can be described by a linear combination of pure states. The difference is that in the non-simplicial theory you have linear combinations of pure states which are indistinguishable by all observations allowed by the theory, and, therefore, identified. In the straightforward "hidden variable" theory they are nonetheless considered to be different states, even if indistinguishable by observation.

However, we know that the laws of physics are governed by a theory with a non-simplicial state space (quantum theory), so it would be unreasonable to apply concepts that only make sense in the context of ordinary probability theory to it.

No. We have decided, based on positivist philosophy, that states which we cannot distinguish by observation in the actual theory (which may be incomplete, but this is nothing a positivist would allow to think about) have to be really identical, and, based on this ideology, rejected the trivial, straightforward extension toward a simplicial probability theory. Or, in other words, we have a philosophical prejudice against hidden variables, formulated in a nice mathematical language, which suggests some deep mathematical insight behind it which in reality does not exist.

And we use this nicely formulated prejudice to reject one of the foundations of the scientific method itself - Reichenbach's principle of common cause, which allows to make a difference between astrology and science.

Sorry if the formulation is a bit too polemically formulated, but I hope you understand the point.

 

[rubi]

No. Ordinary probability theory needs nothing but elementary logic. Read Jaynes, Probability theory: The logic of science.

You can always transform the space of some probability theory into a simplicial one. It is a trivial exercise. Simply postulate that the wave functions (the "pure states" of the non-simplicial theory) are the "hidden variables". That means that you define your "hidden variable" theory has the simplex created by the original pure states.

This is provably wrong. For instance, you cannot possibly model the spin observables $S_i$ as observables on a single probability space. This precludes the formulation of quantum theory on a simplicial state space. (Bohmian mechanics can't do it either.)

No. We have decided

No. You have decided. Scientists have decided differently. Scientists believe that philosophical prejudice has no place in science.

And we use this nicely formulated prejudice to reject one of the foundations of the scientific method itself - Reichenbach's principle of common cause, which allows to make a difference between astrology and science.

Reichenbach tried to formulate mathematically what is common sense about the idea of a common cause. Quantum theory just shows that he failed, since it definitely is common sense to assume that the initial preparation of the system is the common cause for the correlations. That means that the mathematical formulation of our common sense (Reichenbach's principle) needs to be adjusted and not our common sense.

Sorry if the formulation is a bit too polemically formulated, but I hope you understand the point.

Sorry if the formulation is a bit too polemically formulated, but I hope you understand the point.

 

[vanhees71]

However, you do give the false impression that the 100% anti-correlation results (e.g. H vs V) are sufficient to undermine local hidden variables.

That's right, to demonstrate the violation of Bell's inequality you need different angles between A's and B's polarizers than $0$ or $\pi/2$, but within the minimal interpretation that doesn't either need any "spooky action at a distance" to explain the results, because it just says that the state is prepared before any measurement is done and local microcausal QFT tells you that there are only local interactions between the photons and the polarizers at A's and B's position. So the correlations, leading to the violation of Bell's inequality are there from the very beginning when the two photons were prepared and are not caused by A's or B's measurement at the other far distant respective other place.

 

[vanhees71]

I am a bit confused about the mechanics of splitting a photon and the process of parametric down-conversion. My searches only show that the photon passes through a NLO crystal, which seems to be always graphically represented by a box. I am aware of Compton and Thomson scattering in which a photon interacts with a charged particle, but what is happening within the NLO crystal to cause the split? Is it possible that the EM wave is split into its electric and magnetic components and then reacquiring their complimentary components after the split to create two photons of lower energy? But my real interest is in the mechanics of the split in some graphic form similar to how the Compton scattering can be illustrated. Thank you.

Yes, that confuses me for years too. I've not been able to find a microscopic model explaining the parametric down conversion and the preparation of polarization-entangled photon states. What you can find are (pretty simple) phenomenological models assuming the fact that in this way such two-photon states can be prepared (see e.g., [1,2]), but as I said no microscopic models, just using "in-medium QED". Maybe one of the experts here has such a reference?

[1] C. K. Hong, L. Mandel, PRA 31, 2409 (1985)
[2] M. H. Rubin et al, PRA 50, 5122 (1994)

 

[Ilja]

This is provably wrong. For instance, you cannot possibly model the spin observables $S_i$ as observables on a single probability space. This precludes the formulation of quantum theory on a simplicial state space. (Bohmian mechanics can't do it either.)

You have obviously misunderstood the construction.

You have a pure state in quantum theory |\psi\rangle\langle\psi|. You take all these states as the pure states of my HVT, in a single probability space. The resulting simplicial space consists of arbitrary sums (or integrals) of type
\hat{\rho} = \sum_i p_i |\psi_i\rangle\langle\psi_i|
which are in no way different from the states you have in ordinary quantum theory. So the construction does needs nor states not known in standard QT nor something else which does not exist already.

The only difference is that linear combinations which are indistinguishable by observation, because the resulting density operator is the same if the + is interpretated as the sum of operators in Hilbert space, and not as a sum of probability distributions over the basic states |\psi_i\rangle\langle\psi_i|, are, instead, different states of this HVT.

So, nor my construction, nor dBB theory does even try.

No. You have decided. Scientists have decided differently. Scientists believe that philosophical prejudice has no place in science.

Learn to read. My use of "we" was, of course, rhetorical and did not describe my own choice. I thought this would be obvious, just to quote the context:

No. We have decided, based on positivist philosophy, that states which we cannot distinguish by observation in the actual theory (which may be incomplete, but this is nothing a positivist would allow to think about) have to be really identical, and, based on this ideology, rejected the trivial, straightforward extension toward a simplicial probability theory. Or, in other words, we have a philosophical prejudice against hidden variables, formulated in a nice mathematical language, which suggests some deep mathematical insight behind it which in reality does not exist.

But, ok, let's assume you want to tell me my polemics was misguided, and Scientists have decided that such positivistic philosophical prejudice has no place in science. Then your argument disappears into nothing, because if there is no prejudice against theories with hidden variables, there is no base for a prejudice against the construction above, which gives a simplicial probability theory.

Reichenbach tried to formulate mathematically what is common sense about the idea of a common cause. Quantum theory just shows that he failed, since it definitely is common sense to assume that the initial preparation of the system is the common cause for the correlations. That means that the mathematical formulation of our common sense (Reichenbach's principle) needs to be adjusted and not our common sense.

No. It is not common sense to think that violations of Bell's inequality do not prove that there is some hidden communication channel. I agree that it is not our common sense which has to be adjusted, see http://ilja-schmelzer.de/realism/game.php about what I think common sense can tell us about this. But Reichenbach's formulation is fine too.

 

[rubi]

You have obviously misunderstood the construction.

Then you have misunderstood the argument. I'm saying that it is a mathematical theorem that no probability theory on a simplicial state space can reproduce the statistics of three non-commuting spin observables $[S_i, S_j] = i\epsilon_{ijk} S_k$. However, the statistics is experimentally confirmed, so physical descriptions of the situation based on ordinary probability theory are experimentally excluded. Hence, using concepts like Reichenbachs principle, which need ordinary probability theory, are unreasonable in the context of statistics that is incompatible with ordinary probability theory.

No. It is not common sense to think that violations of Bell's inequality do not prove that there is some hidden communication channel. I agree that it is not our common sense which has to be adjusted, see http://ilja-schmelzer.de/realism/game.php about what I think common sense can tell us about this. But Reichenbach's formulation is fine too.

You have explained the common sense of playing cards, which are classical objects. However, photons aren't playing cards, so we can't expect our common sense to apply to them. However, it is common sense to take the preparation of the quantum state to be the cause for the observed correlations. It's your philosophical prejudice that our common sense about playing cards can be carried over to photons.

 

[Ilja]

Then you have misunderstood the argument. I'm saying that it is a mathematical theorem that no probability theory on a simplicial state space can reproduce the statistics of three non-commuting spin observables $[S_i, S_j] = i\epsilon_{ijk} S_k$. However, the statistics is experimentally confirmed, so physical descriptions of the situation based on ordinary probability theory are experimentally excluded.

And I have given a simple counterexample for your claimed mathematical theorem. Which works in a quite general way, for every theory which reproduces whatever statistics. So, I would guess, your "theorem", whatever it is, makes some additional assumptions which exclude this simple and essentially trivial construction.

Again, the trivial construction: Find all the pure states of your theory which reproduces your statistics. Then, define the simplex which has all these pure states as
pure states. Then check that the this simplex reproduces the same statistics. Essentially by construction, because the pure states have the same statistics, and their affine combinations therefore too.

You have explained the common sense of playing cards, which are classical objects. However, photons aren't playing cards, so we can't expect our common sense to apply to them. However, it is common sense to take the preparation of the quantum state to be the cause for the observed correlations. It's your philosophical prejudice that our common sense about playing cards can be carried over to photons.

As well as it is my prejudice that classical logic can be applied everywhere. I'm not ready to accept logical contradictions in a theory simply because the contradictory theory is about some quantum or so things. And probability theory as well as Reichenbach's principle of common cause are, I think, on the same level, see Jaynes. They are the common sense base of science. To modify Kant, science is man's emergence from his self-imposed immaturity. And I'm not ready to accept that somebody else imposes some new Holy or quantum borders behind those it is not allowed to apply common sense, where he has to self-impose his immaturity.

 

[rubi]

And I have given a simple counterexample for your claimed mathematical theorem. Which works in a quite general way, for every theory which reproduces whatever statistics. So, I would guess, your "theorem", whatever it is, makes some additional assumptions which exclude this simple and essentially trivial construction.

Again, the trivial construction: Find all the pure states of your theory which reproduces your statistics. Then, define the simplex which has all these pure states as
pure states. Then check that the this simplex reproduces the same statistics. Essentially by construction, because the pure states have the same statistics, and their affine combinations therefore too.

No, your construction won't recover the statistics of three non-commuting spin observables. It doesn't even work out mathematically, since there are uncountably many pure states, so your sum will diverge. Furthermore, a density matrix doesn't constitute a simplicial state space. The theorem doesn't make any additional assumptions. The only thing you need is the probability distributions that are predicted by QT for non-commuting spin observables. No ordinary probability theory can recover them. It is a fact, which you will have to accept.

As well as it is my prejudice that classical logic can be applied everywhere. I'm not ready to accept logical contradictions in a theory simply because the contradictory theory is about some quantum or so things. And probability theory as well as Reichenbach's principle of common cause are, I think, on the same level, see Jaynes. They are the common sense base of science. To modify Kant, science is man's emergence from his self-imposed immaturity. And I'm not ready to accept that somebody else imposes some new Holy or quantum borders behind those it is not allowed to apply common sense, where he has to self-impose his immaturity.

Quantum theory is logically consistent and the quantum borders are imposed upon us by nature itself, not by any human. Self-imposed immaturity would be to reject mathematical theorems based on ones philosophical prejudices.

 

[Ilja]

No, your construction won't recover the statistics of three non-commuting spin observables. It doesn't even work out mathematically, since there are uncountably many pure states, so your sum will diverge.

Clearly wrong. Learn how to take integrals over probability distributions, say \int \rho(x) dx, over uncountably many real numbers (hint: the result is, in this case, 1).

Furthermore, a density matrix doesn't constitute a simplicial state space.

As if I would have claimed this. It is an element of the quantum state space. If it is a pure density matrix |\psi\rangle\langle\psi|, then we take it and use it as a vertex of some other, simplicial state space. Ok, let's use a different denotation for this element, S_{|\psi\rangle\langle\psi|}. Better?
Then you construct the simplicial state space created by these density matrices as basic states. Each element is a formal linear combination \sum_i p_i S_{|\psi\rangle\langle\psi|}. Feel free to replace the sum (with \sum_i p_i = 1, p_i\ge 0) by an integral (with \int_\lambda \rho(\lambda) d\lambda = 1, \rho(\lambda)\ge 0).

Each element of this simplex then defines a density matrix by a straightforward projection operator \sum_i p_i S_{|\psi\rangle\langle\psi|} \to \sum_i p_i |\psi\rangle\langle\psi|. So for every element of this simplex there exists a corresponding density matrix.

The theorem doesn't make any additional assumptions. The only thing you need is the probability distributions that are predicted by QT for non-commuting spin observables. No ordinary probability theory can recover them. It is a fact, which you will have to accept.

No, I don't have to accept claims which I can easily prove to be wrong, and exist only because you claim it. Give a link to a paper in a peer-reviewed journal where this proof has been made, then we will see what are the additional assumptions.

Self-imposed immaturity would be to believe your claims of existence of some Theorem.

 

[stevendaryl]

I thought it was always due to a conservation law, not just often, are there exceptions?

Well, entanglement doesn't have to involve conservation laws. Any time you have a pair of systems that are in a superposition of states of the form:

|\Psi\rangle = \alpha |\phi_A\rangle |\chi_A\rangle + \beta |\phi_B\rangle |\chi_B\rangle

you have entanglement, and measurement of the first system instantly tells you about the second system, no matter how far away. So the idea of entanglement doesn't have anything to do with conservation laws. However, it might be that in practice, the only way to get such an entangled state is by through conservation laws: Either the total energy, or the total momentum, or the total angular momentum, etc., is fixed, but the partitioning of the quantity between the two systems is different in the two possible elements of the superposition.

 

[rubi]

Clearly wrong. Learn how to take integrals over probability distributions, say \int \rho(x) dx, over uncountably many real numbers (hint: the result is, in this case, 1).

No such integral is defined on the space of operators that you want to sum up.

No, I don't have to accept claims which I can easily prove to be wrong, and exist only because you claim it. Give a link to a paper in a peer-reviewed journal where this proof has been made, then we will see what are the additional assumptions.

Self-imposed immaturity would be to believe your claims of existence of some Theorem.

The proof is really trivial high school mathematics and well-known to every researcher in quantum theory, so there is really no point to doubt it. But here you go:
We try to reproduce the statistics of a quantum state $\left|\Psi\right>=\left|z+\right>$. As everybody knows, the statistics is given by $P(z+)=1$, $P(z-)=0$, $P(x+)=\frac{1}{2}$ and $P(x-)=\frac{1}{2}$.

We want to know whether there is a joint probability distribution $P(s_x,s_z)$ such that $P(x_\pm)=P(\pm,+)+P(\pm,-)$ and $P(z_\pm) = P(+,\pm)+P(-,\pm)$. In particular, we would have $0=P(z-)=P(+,-)+P(-,-)$, i.e. $P(+,-) = - P(-,-)$. Hence, either $P(+,+)$ is negative or $P(-,-)$ is negative or $P(+,+)=P(-,-)=0$.

In the first two cases, we wouldn't have a probability distribution, since probabilities must be positive. In the third case, the system of equations reduces to $1=P(-,+)$, $0=P(+,-)$, $\frac{1}{2}=P(+,-)$ and $\frac{1}{2}=P(-,+)$. Obviously, this system has no solutions ($\frac{1}{2} \neq 1$!), hence no such joint probability distribution exists and hence no observables modeled as random variables on a probability space can reproduce the statistics of the quantum state $\left|\Psi\right>$, since if there were such observables, they would have a well-defined joint probability distribution.

 

[Markus Hanke]

If this would be a valid causal explanation, in agreement with Reichenbach's common cause and Einstein causality, we could prove Bell's theorem.

I disagree. You could prove Bell's theorem only if both realism and Einstein locality hold simultaneously at the time of measurement ( or at least in the period between first interaction and measurement ), implying that there are some form of local hidden variables. Clearly, this is not the case, as evidenced by experiment and observation. The point is that this does not make any reference to, nor does it require, Reichenbach's principle.

On the other hand, the interaction between the two particles in the past demonstrably is the causal explanation of entanglement, because after the interaction has taken place the total information contained in the composite system is greater than the the sum of the information carried by the two subsystems considered in isolation ( I believe the term for this is "subadditivity" ). This would not be the case without the interaction in the past, so causality seems quite obvious to me. The "extra" information contained in the composite system is precisely the correlation implied by entanglement. All of this is of course purely statistical.

Making this mathematically precise isn't so easy for me ( I'm not a scientist ), but my first impulse here would be to write down the reduced density matrices for the two subsystems, and then calculate the entropy from it. The total entropy of the composite system should be less than the sum of the entropies for each particle in isolation, which is a direct causal consequence of their past interaction ( without which we would get an equation, instead of an inequality ).

 

[stevendaryl]

On the other hand, the interaction between the two particles in the past demonstrably is the causal explanation of entanglement, because after the interaction has taken place the total information contained in the composite system is greater than the the sum of the information carried by the two subsystems considered in isolation ( I believe the term for this is "subadditivity" ). This would not be the case without the interaction in the past, so causality seems quite obvious to me. The "extra" information contained in the composite system is precisely the correlation implied by entanglement. All of this is of course purely statistical.

I'm not sure that entanglement necessarily requires interaction in the past, but I don't know of a counter-example.

But the issue with causality is not really about the creation of the entanglement, but about it's consequences.

Alice measures an electron to have spin-up in the z-direction. Because she knows that their particles are entangled, she immediately knows something about Bob's situation--that Bob will not measure spin-up in the z-direction. So think about that statement: "Bob will not measure spin-up in the z-direction". It seems that if the statement is true (in Many-Worlds, it's not true, or no more true than its negation), then we have two possibilities:

1. It became true at some moment.
2. It has always been true.

It's hard to make sense of the first statement without nonlocal effects, and it's hard to make sense of the second statement (in light of Bell's inequality) without superdeterminism.

 

[Ilja]

No such integral is defined on the space of operators that you want to sum up.

You want to tell me that such a space of operators does not allow the definition of a probability measure? Really?

The proof is really trivial high school mathematics and well-known to every researcher in quantum theory, so there is really no point to doubt it. But here you go:
We try to reproduce the statistics of a quantum state $\left|\Psi\right>=\left|z+\right>$. As everybody knows, the statistics is given by $P(z+)=1$, $P(z-)=0$, $P(x+)=\frac{1}{2}$ and $P(x-)=\frac{1}{2}$.

Fine. That means, this state defines a pure state $\left|z+\right>\left<z+\right|$, thus, a vertex of the construction of the simplex.

We want to know whether there is a joint probability distribution $P(s_x,s_z)$ such that ...

And where do you think my construction needs any such "joint probability distribution"? You obviously have not understood the construction at all, invent some meaning of it which requires some "joint probability distribution". Sorry, no. Read again the construction. And recognize that for every state, I repeat every state of the construction there exists an image defined by the map \sum_i p_i S_{|\psi\rangle\langle\psi|} \to \sum_i p_i |\psi\rangle\langle\psi| which is a well-defined density operator of standard QM. I do not have to construct some "joint probability distributions".

Recommended reading: Holevo, Probabilistic and statistical aspects of quantum theory, North Holland, Amsterdam 1982

Theorem 7.1. Any separated statistical model ... is a reduction of a classical model with a restricted class of measurements.

 

[rubi]

And where do you think my construction needs any such "joint probability distribution"?

No, your construction would have to imply the existence of a joint probability distribution, since this existence is automatically guaranteed by probability theory. If I can prove that this joint probability distribution cannot exist, I have also proven that no underlying probability space can exist. You have obviously not understood basic probability theory.

Theorem 7.1. Any separated statistical model ... is a reduction of a classical model with a restricted class of measurements.

This has exactly nothing to do with it.

 

[Ilja]

I disagree. You could prove Bell's theorem only if both realism and Einstein locality hold simultaneously at the time of measurement ( or at least in the period between first interaction and measurement ), implying that there are some form of local hidden variables. Clearly, this is not the case, as evidenced by experiment and observation. The point is that this does not make any reference to, nor does it require, Reichenbach's principle.

May be you know about a variant which needs what you tell us, but does not need Reichenbach's common cause. So what? I have described here already a variant which uses, instead, Reichenbach's common cause together with Einstein causality.

And once there exists such a variant of the proof, that means that once the result - Bell's inequality - is empirically false, or Reichenbach's common cause or Einstein causality is false.

On the other hand, the interaction between the two particles in the past demonstrably is the causal explanation of entanglement, because after the interaction has taken place the total information contained in the composite system is greater than the the sum of the information carried by the two subsystems considered in isolation ( I believe the term for this is "subadditivity" ). This would not be the case without the interaction in the past, so causality seems quite obvious to me. The "extra" information contained in the composite system is precisely the correlation implied by entanglement. All of this is of course purely statistical.

I do not doubt that it is possible to redefine "explanation" so that it becomes possible to interpret this text as an explanation. It is not a common cause explanation in agreement with Reichenbach's common cause principle, because I cannot see any evidence of type P(AB|C) = P(A|C) P(B|C).

 

[Ilja]

No, your construction would have to imply the existence of a joint probability distribution, since this existence is automatically guaranteed by probability theory. If I can prove that this joint probability distribution cannot exist, I have also proven that no underlying probability space can exist. You have obviously not understood basic probability theory.

Given that my construction is a trivial construction equivalent to quantum theory, you have proven that quantum theory does not exist. Congratulations.

This has exactly nothing to do with it.

It has nothing to do with your "no joint probability distribution" argument, indeed. My construction is his, I simply take all the extremal points of the statistical model, in the case of quantum theory the pure states $|\psi\rangle\langle\psi|$, and use them as points in the classical phase space. And it has, in the same way as Holevo's construction, nothing to do with your "no joint probability distribution" argument.

 

[rubi]

Given that my construction is a trivial construction equivalent to quantum theory, you have proven that quantum theory does not exist. Congratulations.

Wrong. I have proven that the statistics of the quantum observables $S_x$ and $S_z$ in the state $\left|\Psi\right>$ cannot be modeled by random variables on a probability space. The argument is watertight, otherwise, you would be able to point out an error, rather than refer to your construction, of which you haven't even attempted to prove that it reproduces said statistics.

It has nothing to do with your "no joint probability distribution" argument, indeed. My construction is his, I simply take all the extremal points of the statistical model, in the case of quantum theory the pure states $|\psi\rangle\langle\psi|$, and use them as points in the classical phase space. And it has, in the same way as Holevo's construction, nothing to do with your "no joint probability distribution" argument.

If you believe that you can reproduce the statistics of the quantum system from my post using a classical probability space, then please just define the objects that are needed, i.e. a probability space $(\Omega, \Sigma)$ with a probability measure $\mu:\Sigma\rightarrow\mathbb R_+$ and the random variables $S_x, S_z :\Omega\rightarrow\left\{-1,1\right\}$ and prove that their probability distributions reproduce the statistics. So far you haven't done this. Just carry out the calculations, please, instead of telling us that it will obviously work out.

Hint: You will not succeed, since I have proven it to be impossible. (If it were possible, I would be able to compute the joint probability distribution of $S_x$ and $S_z$, which I have shown to not exist.)