I How Does Local Measurement Affect an Entangled System?

[nojustay]

TL;DR Summary

I don't understand entanglement

Hi, I am doing my thesis on quantum entanglement and I don't seem to wrap my head around what really happens to an entangled system during a local measurement. I happen to know that information can't travel faster than light I could believe that the collapse of the wave function wouldn't allow us to really extract any relevant data. Yet, the particles themself "know" about the collapse and act accordingly. For example, the decay charmonium -> K_s K_s never happens even in space separated regions.

 

[DrChinese]

Quantum entanglement is a big field. Do you have a specific question?

You mention doing a thesis on entanglement. I would recommend you read up on standard photon entanglement theory and experiments. Most of these use Parametric Down Conversion to create and study entanglement. Exotic particles (such as charmonium) are not a good place to start.

 

[Herman Trivilino]

TL;DR Summary: I don't understand entanglement

Yet, the particles themself "know" about the collapse and act accordingly.

Collapse is just one of many interpretations. The particles don't act accordingly. That's a misconception based on classical reasoning.

 

[Meir Achuz]

"I happen to know that information can't travel faster than light"
That doesn't apply in nonrelativistic quantum mechanics.

 

[pines-demon]

For example, the decay charmonium -> K_s K_s never happens even in space separated regions.

I am interested in that charmonium decay. Do you have any source on that?

 

[Sargon38]

TL;DR Summary: I don't understand entanglement

Hi, I am doing my thesis on quantum entanglement and I don't seem to wrap my head around what really happens to an entangled system during a local measurement. I happen to know that information can't travel faster than light I could believe that the collapse of the wave function wouldn't allow us to really extract any relevant data. Yet, the particles themself "know" about the collapse and act accordingly. For example, the decay charmonium -> K_s K_s never happens even in space separated regions.

My take with entanglement is an MWI like view. But that's just interpretational, to give a mental picture to "what happens".

Entanglement is just the phenomenon that non-product states of compound systems generate different correlations between the subsystem observed probabilities according to the basis in which one looks at the subsystems (that is to say, according to the different incompatible measurements one actually does to the subsystems).

If you don't want to go "MWI" you can't really attach much "physical reality" to wavefunctions: you have to see it as a device that generates probabilities for measurements.

If you have a state $|A_1> \otimes |A_2> + |B_1> \otimes |B_2>$ then you would get, if you measure systems 1 and 2 in the basis $\{ |A>, |B> \}$ probabilities of 1/2 for (A,A) and 1/2 for (B,B) outcomes, and probability 0 for outcomes (A,B) and outcomes (B,A).
Up to now, there's no distinction between a classical mixture with these probabilities, and this quantum state. There's no particular "indication" of entanglement. You get a classical correlation, where system 1 has 50% chance to be in state A, and 50% chance to be in state B, and system 2 is perfectly correlated with system 1.

The weird things start to happen when you measure system 1 and system 2 in different bases. Your entangled state will still provide you with the probabilities of the combined outcomes, and these probabilities will still be "Kolmogorov standard probabilities". However, what fails is to assume that the outcomes of contrafactual measurements are still described by Kolmogorov probabilities. (that's essentially Bell's theorem).

Nevertheless, once you have determined your measurements (the bases you will use), the quantum entangled state will spit out the correct set of Kolmogorov probabilities for the combinations of outcomes.

The essence of entanglement is that you can't have a Kolmogorov probability set for contrafactual measurements a priori.

 

[DrChinese]

1. The weird things start to happen when you measure system 1 and system 2 in different bases. Your entangled state will still provide you with the probabilities of the combined outcomes, and these probabilities will still be "Kolmogorov standard probabilities".

2. However, what fails is to assume that the outcomes of contrafactual measurements are still described by Kolmogorov probabilities. (that's essentially Bell's theorem). ... The essence of entanglement is that you can't have a Kolmogorov probability set for contrafactual measurements a priori.

Your description of entanglement oversimplifies in a manner that IMHO won't help the general reader.

1. When you measure on the same bases, there are perfect correlations (a la EPR). And when you measure on different bases, there is nothing about "Kolmogorov probabilities" that enters the picture at all. QM is contextual, and the Kolmogorov probabilities you are referring to only come into play when you consider counterfactual reasoning associated with non-contextuality (realism) - if they are to be considered at all.

The quantum prediction for entangled systems is strictly statistically related to those bases (same or different). As far as is currently known, there is no other variable (or set thereof) that determines or influences the expected results in any way. Only a future context matters, regardless of the size (separation) of the entangled system.

2. I would never describe Kolmogorov probabilities as having anything to do with the essence of entanglement. You mentioned in your post (to the effect of) that an entangled system cannot be considered as 2 individual systems, since by definition entangled systems cannot be represented by a Product of 2 systems. Agreed, that is more the essence of entanglement.

Likewise, I wouldn't say Kolmogorov probabilities have much to do with Bell. It is the assumptions of locality and realism (counterfactuals) that constitute Bell. If you want to say Bell also assumes Kolmogorov probabilities, then fine. I would say those are also assumed for most of science, nothing particular about Bell. All possibilities must add to 100% when counterfactuals are possible - which of course does not always apply in QM (as Bell demonstrated).

 

[Sargon38]

Your description of entanglement oversimplifies in a manner that IMHO won't help the general reader.

1. When you measure on the same bases, there are perfect correlations (a la EPR). And when you measure on different bases, there is nothing about "Kolmogorov probabilities" that enters the picture at all. QM is contextual, and the Kolmogorov probabilities you are referring to only come into play when you consider counterfactual reasoning associated with non-contextuality (realism) - if they are to be considered at all.

My point was: when one doesn't change bases then there's strictly no difference between "classical" (Kolmogorov) correlations and entanglement. The "weirdness of entanglement" only shows if one changes bases, and one is going to perform incompatible measurements. That's, I suppose, what you mean by "contextual".
For instance, if one has an entangled spin state of two particles, and one keeps measuring particle 1's spin in, say, the X direction, and particle 2's spin state in the Z direction, then this can perfectly be described by a classical correlation between the X spin result of particle 1 and the Z spin result of particle 2, which is the set of probabilities one obtains when projecting the entangled state in the basis of $X \otimes Z$. The projections provide you with "normal" Kolmogorov-type probabilities.

It is only when you "change basis" and, say, analyse this system ALSO in the $Y \otimes X$ basis, say, that you WOULD get "weird, non-Kolmogorov" type probabilities (violating Bell's inequalities) if you assumed counterfactually that the previously calculated probabilities in the other basis were still valid too. But this is counterfactual, because you cannot do the measurements *simultaneously* in both bases.

But it is the appearence of these "counterfactual" non-Kolmogorov probabilities that characterize entanglement as opposed to "statistical mixture". It is what separates the entangled state from the reduced density operator.

As long as you don't consider the system in different bases, you can't make the difference. That was my point: that it is this "weirdness" that appears in true entangled superpositions, which is absent in statistical mixtures, which is the essence of entanglement.

It is what makes quantum mechanics fundamentally different from just statistical mixtures, but it only appears in counterfactual situations. As you said, any actual measurement will always be satisfying normal Kolmogorov probabilities, derivable from the quantum state in the basis of observation.

The quantum prediction for entangled systems is strictly statistically related to those bases (same or different). As far as is currently known, there is no other variable (or set thereof) that determines or influences the expected results in any way. Only a future context matters, regardless of the size (separation) of the entangled system.

It is exactly this absence of the possibility of a (normal hence Kolmogorov) probability distribution of hidden variables that characterizes entanglement. That was my whole point.

2. I would never describe Kolmogorov probabilities as having anything to do with the essence of entanglement. You mentioned in your post (to the effect of) that an entangled system cannot be considered as 2 individual systems, since by definition entangled systems cannot be represented by a Product of 2 systems. Agreed, that is more the essence of entanglement.

Well, it is exactly the impossibility of Kolmogorov probabilities for all possible outcomes of measurements on entangled states that characterizes entangled states, that was my point.

Likewise, I wouldn't say Kolmogorov probabilities have much to do with Bell. It is the assumptions of locality and realism (counterfactuals) that constitute Bell. If you want to say Bell also assumes Kolmogorov probabilities, then fine. I would say those are also assumed for most of science, nothing particular about Bell. All possibilities must add to 100% when counterfactuals are possible - which of course does not always apply in QM (as Bell demonstrated).

That is a strange statement, because Bell's inequalities are based upon standard (Kolmogorov) probability theory, and Bell's theorem is exactly the statement that any objective a priori description of individual subsystem states in an entangled system that would yield the measurement outcomes would need probabilities that violate these inequalities.

So to me, normal Kolmogorov probabilities are central to what Bell expressed, and what Bell expresses is probably the most characteristic aspect of entanglement.

 

[PeroK]

My point was: when one doesn't change bases then there's strictly no difference between "classical" (Kolmogorov) correlations and entanglement.

Bell's Theorem was the death knell for local hidden variables and entanglement and tests of Bell's theorem provided a great experimental result for QM. The underlying principle of QM of using (complex) probability amplitudes, as opposed to classical (Kolmogorov) probabilities, was already well established. To explain the double-slit experiment, the Stern-Gerlach experiment etc. Moreover, particle scattering calculations explicitly calculate a probability amplitude first.

In fact, the popular wave-particle explanation for the double-slit plays on the lay person's need to think in terms of concrete probabilities: the electron is definitely a wave; then it is definitely a particle etc. The correct QM calculation that involves complex probability amplitudes is side-lined.

In summary, even without entanglement, you cannot maintain an interpretation of QM that relies on classical probabilities at each stage.

 

[DrChinese]

My point was: when one doesn't change bases then there's strictly no difference between "classical" (Kolmogorov) correlations and entanglement. The "weirdness of entanglement" only shows if one changes bases, and one is going to perform incompatible measurements. That's, I suppose, what you mean by "contextual".

...

So to me, normal Kolmogorov probabilities are central to what Bell expressed, and what Bell expresses is probably the most characteristic aspect of entanglement.

The issues about entanglement theory per QM have nothing to do with the classical world. The OP is trying to understand what entanglement is, not what it isn't - "I don't understand entanglement". I think it is just as amazing that perfect correlations exist, because that might imply that (as EPR claimed) QM is incomplete - when Bell shows it is not.

Again, I have no idea why you are focused on Kolmogorov. That is virtually never discussed in treatments of entanglement or Bell, mainly it is discussed by Bell deniers (which you are not). If you can even find a reference in the Bell 1964 paper that remotely makes this a relevant topic to this thread, I'd be interested to see the quote.

At any rate, there is far more to entanglement than Bell (although I think it is mandatory for any hope of understanding entanglement). 35 years ago we got the GHZ Theorem. In that, there are no statistical issues (as in Bell) that might remotely relate to Kolmogorov ideas. Every single outcome of every trial is exactly the opposite of what a classical perspective would predict. And of course, in perfect agreement with the predictions of QM.

 

[DrChinese]

So to me, normal Kolmogorov probabilities are central to what Bell expressed, and what Bell expresses is probably the most characteristic aspect of entanglement.

PS Not that it proves anything*, but I searched for "Kolmogorov" in Stanford's Plato for these pages:

Bell's Theorem
Quantum Entanglement and Information
Philosophical Issues on Quantum Theory

Total hits: 0. There isn't even a page for Kolmogorov.


*Well, actually it does...

 

[PeroK]

PS Not that it proves anything*, but I searched for "Kolmogorov" in Stanford's Plato for these pages:

Bell's Theorem
Quantum Entanglement and Information
Philosophical Issues on Quantum Theory

Total hits: 0. There isn't even a page for Kolmogorov.


*Well, actually it does...

To be fair, Kolmogorov is just another name for classical probability theory.

 

[Sargon38]

Bell's Theorem was the death knell for local hidden variables and entanglement and tests of Bell's theorem provided a great experimental result for QM. The underlying principle of QM of using (complex) probability amplitudes, as opposed to classical (Kolmogorov) probabilities, was already well established.

Yes, but the whole thing about Bell was exactly that two different subsystems (with the potential of being spatially remote) could be in a state that was NOT to be described by a statistical mixture. That was the exact content of Bell's theorem, and to me, the essence of what one calls entanglement.
Entanglement as distinguished from statistical mixture. It is exactly the feature of not being describable by (counterfactual) Kolmogorov probabilities that renders entanglement different from statistical mixture (by definition I would say). If all measurable properties of subsystems were describable by a statistical mixture, there wouldn't be anything special about "entanglement". It would have been just a fancy way to express statistical correlations. It is exactly because this is NOT possible that entanglement "stands out" as, well, entanglement.

But, as I said earlier, that is of course only visible if one uses counterfactual situations, because quantum mechanics ALWAYS delivers "normal probabilities" for a given realisable measurement. So in order to see the particularity of entanglement, namely that it is NOT just "statistical mixture", it is essential to look to the state (mentally) in non-compatible bases (exactly to obtain counterfactual situations).

To me that is exactly what entanglement (as opposed to statistical mixture) is about.

 

[Sargon38]

To be fair, Kolmogorov is just another name for classical probability theory.

That's what I mean by "Kolmogorov", to distinguish this from some other, weird, "quantum probability" concept. I mean by Kolmogorov probabilities, standard, textbook, statistician, probabilities.
Sorry for the confusion, I thought that it would rather remove them.

 

[PeroK]

Yes, but the whole thing about Bell was exactly that two different subsystems (with the potential of being spatially remote) could be in a state that was NOT to be described by a statistical mixture.

States in QM are generally not statistical in the classical (Kolmogorov) sense, but statistical in the sense of having complex probability amplitudes.

 

[DrChinese]

To be fair, Kolmogorov is just another name for classical probability theory.

Agreed and that's why I said it's common for most areas of science. "those are also assumed for most of science, nothing particular about Bell."

 

[PeterDonis]

the whole thing about Bell was exactly that two different subsystems (with the potential of being spatially remote) could be in a state that was NOT to be described by a statistical mixture. That was the exact content of Bell's theorem

No, it isn't. Bell's theorem itself has nothing whatever to do with states that can't be described by statistical mixtures. Bell's theorem itself is entirely about what kinds of correlations can be produced by states that can be described by a statistical mixture. More generally, theorems like Bell, GHZ, etc. are about what kinds of predictions can be produced by models that satisfy the kinds of assumptions that you are referring to when you use terms like "statistical mixture" (although, as @DrChinese has said, in cases like GHZ the predictions aren't statistical at all).

It is the fact that the predictions of QM, and now many experimental results confirming those predictions, are inconsistent with the predictions that can be produced by models that satisfy the conditions of the above theorems, that you are referring to by the term "entanglement". But that very fact invalidates any discussion of the actual QM predictions, and the actual experimental facts, in terms that make the kinds of assumptions described above. The implication of the actual experimental facts is that the real world does not satisfy those assumptions.

 

[PeterDonis]

that is of course only visible if one uses counterfactual situations

This is not correct. The fact that QM predictions and the actual experimental facts violate the Bell inequalities, the GHZ predictions, etc. can be, and has been, confirmed without any resort to counterfactuals at all. The predictions are about what we will actually observe when we do actual experiments; there is no need at all to drag in claims about what we would have observed if we had done different experiments.

 

[DrChinese]

That's what I mean by "Kolmogorov", to distinguish this from some other, weird, "quantum probability" concept. I mean by Kolmogorov probabilities, standard, textbook, statistician, probabilities.
Sorry for the confusion, I thought that it would rather remove them.

Quantum predictions are what matters in discussions of entanglement. Are you saying those are weird? What one person labels "weird" might be quite different than another.

 

[Sargon38]

No, it isn't. Bell's theorem itself has nothing whatever to do with states that can't be described by statistical mixtures. Bell's theorem itself is entirely about what kinds of correlations can be produced by states that can be described by a statistical mixture.

Ah, I thought that Bell's inequalities are what you are referring to, and that Bells' theorem was that certain entangled quantum states violate these inequalities if we produce counterfactual probabilities and correlations when looking in different non-compatible measurement bases.

Now, I should look it up again. Bell's inequalities are inequalities that all classical (Kolmogorov) correlations must satisfy. The (counterfactual, because calculated in different bases) correlations that one obtains from certain entangled states violate these inequalities, and I thought that this was what Bell's theorem was about.

Maybe some people call Bell's inequalities "Bell's theorem", I didn't remember it that way.

My whole point in this thread was, that it is exactly THIS violation of the possibility of describing the system by a statistical mixture (which is exactly what the violations of Bell's inequality are about) are the VERY ESSENCE of what one calls "entanglement" (which was the OP question).

If it wasn't for such impossibility to describe an entangled state by a statistical mixture, then entanglement wouldn't be anything particular. It would be a fancy way to say that we have simply a statistical mixture of "couples of outcomes". It is because exactly that is not possible, that entanglement is what it is.

It is the fact that the predictions of QM, and now many experimental results confirming those predictions, are inconsistent with the predictions that can be produced by models that satisfy the conditions of the above theorems, that you are referring to by the term "entanglement".

Yes, that's what I'm saying from the beginning, that is EXACTLY what entanglement is. Otherwise it would be simply a statistical mixture, and there wouldn't be anything fancy about "entanglement".

I'm having myself a counter factual observation in this thread, that I'm saying "what makes entanglement what it is, is exactly the non-existence of counterfactual classical (Kolmogorov) probabilities of all (counterfactual) outcomes" and I get a list of people saying "no no no, you are wrong, in quantum mechanics, there are no classical counterfactual probabilities ! "

 

[DrChinese]

Yes, but the whole thing about Bell was exactly that two different subsystems (with the potential of being spatially remote) could be in a state that was NOT to be described by a statistical mixture. That was the exact content of Bell's theorem, and to me, the essence of what one calls entanglement.
Entanglement as distinguished from statistical mixture. It is exactly the feature of not being describable by (counterfactual) Kolmogorov probabilities that renders entanglement different from statistical mixture (by definition I would say). If all measurable properties of subsystems were describable by a statistical mixture, there wouldn't be anything special about "entanglement". It would have been just a fancy way to express statistical correlations. It is exactly because this is NOT possible that entanglement "stands out" as, well, entanglement.

But, as I said earlier, that is of course only visible if one uses counterfactual situations, because quantum mechanics ALWAYS delivers "normal probabilities" for a given realisable measurement. So in order to see the particularity of entanglement, namely that it is NOT just "statistical mixture", it is essential to look to the state (mentally) in non-compatible bases (exactly to obtain counterfactual situations).

To me that is exactly what entanglement (as opposed to statistical mixture) is about.

Your usage of the term "statistical mixture" does not provide meaning here, it is another confusing usage. The normal terms related to entangled vs separable (non-entanglement) systems are "Entangled State" statistics and "Product State" statistics. It does not make sense to use the word Kolmogorov in any manner here at all. You may as well talk about the assumption that 1+0=1. Not relevant.

Similarly, the usage of the phrase "normal probabilities" is confusing at best. QM makes predictions that can be tested, not sure what is normal or not about them. It is possible you are referring to "normalizable" or similar, but again that has no bearing at all.

There are plenty of interesting issues around entanglement, and the one you seem to be focused on is pretty low in any discussion on the subject I've seen. How about quantum nonlocality? I would say that is pretty high - and certainly more interesting. That's certain to generate debate amongst followers of different interpretations (which of course should only debated in the related PF subforum).

 

[PeroK]

I'm having myself a counter factual observation in this thread, that I'm saying "what makes entanglement what it is, is exactly the non-existence of counterfactual classical (Kolmogorov) probabilities of all (counterfactual) outcomes" and I get a list of people saying "no no no, you are wrong, in quantum mechanics, there are no classical counterfactual probabilities ! "

What I'm saying is that even without considering entanglement, we have non-classical QM with non-classical probability theory. It doesn't need entanglement to make nature quantum mechanical.

 

[Sargon38]

This is not correct. The fact that QM predictions and the actual experimental facts violate the Bell inequalities, the GHZ predictions, etc. can be, and has been, confirmed without any resort to counterfactuals at all. The predictions are about what we will actually observe when we do actual experiments; there is no need at all to drag in claims about what we would have observed if we had done different experiments.

The Bell inequalities only hold assuming a common single probability universe for outcomes that are counterfactual, that's the whole point of it. If you do not consider counterfactual situations in the same probability universe, then you cannot derive Bell's inequalities. It are not the real world experiments that are counter factual (of course !), it are the hypotheses you have to make to derive Bell's inequalities.

In order for Bell's inequalities to hold, you have to consider that there is a single probability distribution over a universe that contains AS WELL the +X outcome, as the +Z outcome of particle 1, say. That's counterfactual: you can't measure AS WELL the +X outcome as the +Z outcome. It is ASSUMING that you could, that you can derive Bell's inequalities concerning the correlations of outcomes. This "single universe" with its single probability distribution is materialized by the probability distribution assumed over the "hidden variables".
It is this which is counterfactual.

 

[DrChinese]

1. Ah, I thought that Bell's inequalities are what you are referring to, and that Bells' theorem was that certain entangled quantum states violate these inequalities if we produce [assume] counterfactual probabilities and correlations when looking in different non-compatible measurement bases.

2. Bell's inequalities are inequalities that all classical (Kolmogorov) correlations must satisfy.

3. I'm having myself a counter factual observation in this thread, that I'm saying "what makes entanglement what it is, is exactly the non-existence of counterfactual classical (Kolmogorov) probabilities of all (counterfactual) outcomes" and I get a list of people saying "no no no, you are wrong, in quantum mechanics, there are no classical counterfactual probabilities ! "

1. As revised, this is basically accurate. Locality must also be assumed. Bell's Theorem in a nutshell:

"No physical theory of local Hidden Variables can ever reproduce all of the predictions of Quantum Mechanics."

2. Notice that the above has nothing to do with probability theory. It's no more complicated than: 1/3 + 1/3 + 1/3 = 100%. No need to throw poor Kolmogorov's name around.

3. As it relates to Bell, QM does not have anything to say about counterfactuals at all. Only those who believe that QM is incomplete (in the EPR sense) care about those. EPR specifically discusses counterfactuals in their conclusions.

 

[PeterDonis]

Bells' theorem was that certain entangled quantum states violate these inequalities

No, that's not what Bells' theorem says. Bell's theorem is that any model that satisfies certain assumptions must make predictions that satisfy certain inequalities (the Bell inequalities). The fact that QM's predictions violate those inequalities means that QM, as a model, cannot satisfy the assumptions of Bell's theorem. And since QM's predictions have been experimentally confirmed, the real world itself cannot satisfy the assumptions of Bell's theorem.

it is exactly THIS violation of the possibility of describing the system by a statistical mixture (which is exactly what the violations of Bell's inequality are about) are the VERY ESSENCE of what one calls "entanglement"

It's not just a matter of statistics, as has already been said. Bell's theorem itself was about statistical predictions, but other theorems of the same general class, such as GHZ, are not. So just saying "we can't describe actual entangled quantum systems in the real world by statistical mixtures" does not fully convey the facts.

The Bell inequalities only hold assuming a common single probability universe for outcomes that are counterfactual

Even if this is the case for Bell's theorem (I believe the literature generally refers to what you are describing in the assumptions of that theorem as "counterfactual definiteness"), it is not the case for other theorems in the same class, such as GHZ. The GHZ predictions are not about statistics but about scenarios where you make a binary measurement, the "classical" model says you get one result, but QM (and actual experiment) says you get the other. No statistics, no counterfactuals. Just a straight yes/no test.

 

[Sargon38]

What I'm saying is that even without considering entanglement, we have non-classical QM with non-classical probability theory. It doesn't need entanglement to make nature quantum mechanical.

Quantum mechanics always spits out Kolmogorov probabilities for all measurable quantities. QM never produces "non-classical" probabilities in non-counterfactual situations. You always get "normal" probabilities over any complete set of compatible observables.

But my point from the beginning was what sets out ENTANGLEMENT from a statistical mixture of the sub systems, is exactly that you get non-standard probabilities (or call it, broken probabilities) if you DO consider counterfactual measurements on the subsystems. If not, there wouldn't be any point in considering entanglement in the first place: we could just as well study simply statistical mixtures of product states.
So it is the very existence of these counterfactual violations of "standard probabilities" that set entangled states apart from statistical mixtures.

That was my answer to "what is enganglement".

 

[DrChinese]

The Bell inequalities only hold assuming a common single probability universe for outcomes that are counterfactual, that's the whole point of it. If you do not consider counterfactual situations in the same probability universe, then you cannot derive Bell's inequalities. It are not the real world experiments that are counter factual (of course !), it are the hypotheses you have to make to derive Bell's inequalities.

This is not how the historical derivation of counterfactuals occurred. It went from EPR (1935) to Bell's counterargument (1964). It was EPR that raised the idea of counterfactuals, and they specifically mention that they would not have achieved their conclusion (QM is incomplete) without assuming them. Bell is an attack on EPR, not on QM. Of course, it could have led to the discovery that QM was wrong too.

You are trying to pick out one element of the Bell proof as if it is the only one that matters. And it is probably the least interesting element (that classic probabilities sum to 100%). Not what I would call a burning issue.

 

[DrChinese]

Quantum mechanics always spits out Kolmogorov probabilities for all measurable quantities.

This is literally gibberish wrong. For entangled photon polarization experiments, the QM match prediction is cos^2(A-B). Nothing to do with Kolmogorov, standard textbook probability theory, or anything else that you have mentioned.

 

[PeroK]

Quantum mechanics always spits out Kolmogorov probabilities for all measurable quantities. QM never produces "non-classical" probabilities in non-counterfactual situations. You always get "normal" probabilities over any complete set of compatible observables.

But my point from the beginning was what sets out ENTANGLEMENT from a statistical mixture of the sub systems, is exactly that you get non-standard probabilities (or call it, broken probabilities) if you DO consider counterfactual measurements on the subsystems. If not, there wouldn't be any point in considering entanglement in the first place: we could just as well study simply statistical mixtures of product states.
So it is the very existence of these counterfactual violations of "standard probabilities" that set entangled states apart from statistical mixtures.

That was my answer to "what is enganglement".

I see what you mean. It was the emphasis on probabilites, rather than correlations that I didn't follow. I guess it amounts to the same thing.

PS although I'm not convinced that the Kolmogorov axioms themselves are violated. Because, at the end of the day, the axioms of probability theory don't know what physical assumptions are allowed - like non-locality. I'm not convinced that physical non-locality breaks probability theory. And, if it does, which axioms fail?

I think it's better to say that the correlations predicted by QM cannot be sustained given the physical assumptions of Bell's Theorem.

 

[PeterDonis]

Kolmogorov probabilities

"normal" probabilities

non-standard probabilities (or call it, broken probabilities)

"standard probabilities"

If you want to continue using these terms, you're going to have to give actual definitions of them, and back those definitions up with references. This appears to me to be your personal idiosyncratic terminology. That's not acceptable here.

 

[Sargon38]

This is literally gibberish wrong. For entangled photon polarization experiments, the QM match prediction is cos^2(A-B). Nothing to do with Kolmogorov, standard textbook probability theory, or anything else that you have mentioned.

In what way would $\cos^2 (\theta_A - \theta_B)$ not be a probability distribution that satisfies the standard (Kolmogorov) properties of probability theory ?
What's so impossible to the statement "if you put your polarizers $\theta$ degrees apart, you're going to measure a correlation in clicks equal to $\cos^2(\theta)$ ? Would a statistician yell at you "impossible !!" ?

 

[PeroK]

PPS this is why the superdeterminists can try to get round Bell's Theorem by invoking correlations between the state to be measured and the measuring apparatus. The Kolmogorov axioms themselves are not violated in that case. It's that you need alternative physical assumptions to explain the correlations.

 

[PeroK]

This is literally gibberish wrong. For entangled photon polarization experiments, the QM match prediction is cos^2(A-B). Nothing to do with Kolmogorov, standard textbook probability theory, or anything else that you have mentioned.

I don't think this is right. I'm not convinced that QM breaks probability theory. If it does, you'd have to specify which of the Kolmogorov axioms fail. The axioms themselves demand no specific physical context.

 

[Sargon38]

If you want to continue using these terms, you're going to have to give actual definitions of them, and back those definitions up with references. This appears to me to be your personal idiosyncratic terminology. That's not acceptable here.

I thought that the Kolmogorov axioms of a probability distribution over a universe $\Omega$ were sufficiently standard.
I mean standard schoolbook probability theory as it is usually taught in any course on probability theory.
This in order to distinguish this from any funny variants such as "quantum probabilities" and the like.

 

[PeroK]

I thought that the Kolmogorov axioms of a probability distribution were sufficiently standard.
I mean standard schoolbook probability theory as it is usually taught in any course on probability theory.
This in order to distinguish this from any funny variants such as "quantum probabilities" and the like.

Here they are:

https://en.wikipedia.org/wiki/Probability_axioms

 

[PeterDonis]

This in order to distinguish this from any funny variants such as "quantum probabilities" and the like.

Again, you need to back up terms like this with definitions and references. Yes, we know what the Kolmogorov axioms are, but that isn't the only term you've been using.

 

[DrChinese]

I don't think this is right. I'm not convinced that QM breaks probability theory. If it does, you'd have to specify which of the Kolmogorov axioms fail. The axioms themselves demand no specific physical context.

There is NO application of the Kolmogorov axioms in the Entangled State statistics; and nowhere do I say QM "breaks probability theory" (although it does in special cases, but that's a different and complex subject altogether). Again, these have NO purpose being brought up here. Someone start a thread on the Kolmogorov axioms and we can discuss there. But his thread is subtitled: "I don't understand entanglement". How dos textbook probability theory - under a fancy name intended to obscure - apply to the cos^2 formula? QM gives a prediction, and you don't need any other probability rules to get that.

The [Kolmogorov] axioms themselves demand no specific physical context. True. So what does that have to do with entanglement? Or separable states? It's just a random statement that is so important to Entanglement, no one bothers to include it in summaries of the subject.

 

[DrChinese]

What's so impossible to the statement "if you put your polarizers $\theta$ degrees apart, you're going to measure a correlation in clicks equal to $\cos^2(\theta)$ ? Would a statistician yell at you "impossible !!" ?

The statistician certainly wouldn't bother to say: It will be a number between 0% and 100% inclusive. It would be a true statement, but as I said, completely meaningless and of no value in a discussion. There are lots of things that are true, that need not be mentioned in a PF thread. Particularly this one. As I have said: if it's so important, start a new thread.

 

[PeroK]

The [Kolmogorov] axioms themselves demand no specific physical context. True. So what does that have to do with entanglement?

The Kolmogorov axioms are an implicit assumption in Bell's inequality - hence have some relevance to entanglement - and specifically why local hidden variables are ruled out.

The correlations predicted by QM, however, can be explained by adopting a different set of physical assumptions. There is nothing special about them in the context of pure probability theory. The specific probabilities predicted by QM do not "break probability theory" in any sense.

 

[DrChinese]

The Kolmogorov axioms are an implicit assumption in Bell's inequality - hence have some relevance to entanglement - and specifically why local hidden variables are ruled out.

The correlations predicted by QM, however, can be explained by adopting a different set of physical assumptions. There is nothing special about them in the context of pure probability theory. The specific probabilities predicted by QM do not "break probability theory" in any sense.

Again, there are lots of things that are implicit in any area of science - they all don't need mentioning. Again, explain why they are so important that they aren't mentioned in discussions of Entanglement (other than this thread) ? So no, why claim relevance of something that is assumed by scientists (and statisticians) everywhere?

No respectable scientist would mention Kolmogorov in a published discussion (summary) of entanglement. Any more than anyone would talk about standard probability theory. And why use a buzzword except to obscure?

And so what if it has some tangential relationship to Bell? This thread asks "what is entanglement?" Does Sargon38's line of discussion of any purpose at all? In my opinion, using the word "Kolmogorov" in this thread should be banned. Of course, you can't thread ban a word.

 

[Sargon38]

Again, you need to back up terms like this with definitions and references. Yes, we know what the Kolmogorov axioms are, but that isn't the only term you've been using.

Well, there are equivalent terms like "standard" probabilities, "classical" probabilities and so on because I had the impression people got nervous at the term Kolmogorov. The problem with using "classical" is that this might be mis-interpreted as being something related to "classical physics" or so, where I only wanted to indicate "the usual KIND of probability theory" (which is why "Kolmogorov" seemed more to the point to me, but this had some enerving effect on Dr Chinese :-) ). And "standard" might also be mis-interpreted as being something else than "the probabilities satisfying the standard properties of a probability distribution over a universe".

It seemed to me that people were objecting to things I didn't say, and that this was a kind of mis understanding. In the case of misunderstandings, one tries to reformulate with different words in order for the meaning to be conveyed. If one is going to play semantic games when one tries to do so, of course, it is not difficult to set up a straw man, because the more I'm going to try to say things differently in order to lift misunderstandings, the more I will get questions and remarks over "you didn't define those terms".

I try once more to reformulate what I was saying in this thread: the *characteristic* of entangled states is that they imply "probability distributions" (and specifically under the form of correlations) which are NOT Kolmogorov under non-contextual hypotheses (usually called hidden variable hypotheses). I put "probability distributions" in quotes because their Kolmogorov-violating aspects make that it aren't probability distributions.

This is to me what is the essence of entanglement, and what distinguishes it fundamentally from a statistical mixture, where, of course, by definition, the generated probability distribution (and hence the correlations that are part of it) would satisfy the Kolmogorov axioms.

BTW, this discussion triggered a question I have myself: do we get such non-Kolmogorov counterfactual probabilities also in the case of "simple" quantum states which we usually don't call "entangled" ? I'm inclined to think "no", but I'm actually not sure.

Essentially, what we have is:

given (pure) quantum state + given observation basis => "standard" Kolmogorov probability distribution over the universe of possible outcomes (in that basis of course).

That was the point I was making, that quantum mechanics ALWAYS generates "standard" probabilities over actual possible measurement outcome universes. There nothing non-Kolmogorov to quantum theory generated probabilities that can be done in reality. These are standard, classical, probabilities over the possible outcomes, and if the outcomes are multi-valued, I understand by probabilities of course also all the correlations between those values.

When one takes the same quantum state, but one changes the observation basis, we get a DIFFERENT probability distribution over ANOTHER universe of possible outcomes. These are ALSO standard probabilities, but as we have now a totally different set of outcomes, there's no link with the previous universe.

What was characteristic to "entanglement" in my mind was that the quantum system consists of "separated sub systems" over which we can do different measurements. Now, there's still nothing special about this: if we pick a given observation basis, we STILL find a standard probability distribution over the outcome universe of this combined system. And if we use a DIFFERENT observation basis (and the same pure entangled state) we get yet ANOTHER standard probability distribution over the different outcome universe of this combined system.

However, and that was my point, as we have different subsystems, this time we may keep the same observations on one subsystem, and then change the measurement on the other subsystem. We can keep measuring the Z spin on particle one, and we can change the spin measurement to X on particle 2.

And by doing this, we have of course individual pair-wise correlations that have been derived from DIFFERENT probability distributions generated by quantum theory between the outcomes of measurement on system 1 and on system 2.

Well, the pecularity is that these 2-by-2 correlations that are part of different (but still standard) probability distributions are NOT derivable from a single probability distribution on a counterfactual "master" universe of observations that would encompass the different previously introduced different measurement universes.

Again, in a way, this shouldn't surprise us, because we have been mixing up correlations that were part of DIFFERENT probability distributions over DIFFERENT measurement outcome universes. But it shows that the "counterfactual universe of observations and its corresponding probability distribution" doesn't exist, which a physicist calls "local hidden variables".

But it is in my eyes essential, because if it weren't, then all entangled states could be seen as equivalent to statistical mixtures over this "counterfactual universe of observations". It is exactly because they can't that they have something special about them, and that's why I wanted to say that THIS is what "characterises" entanglement.

Now to come back to my question: does this ONLY happen in states that we call "entangled" (my intuition thinks so) or is this also to be found in more down-to-earth simpler quantum systems that one doesn't call entangled ? Say, the single scalar particle or so ?

Because it was based on my "intuition" that I called that aspect as TYPICAL for "entanglement".

 

[Sargon38]

Does Sargon38's line of discussion of any purpose at all? In my opinion, using the word "Kolmogorov" in this thread should be banned. Of course, you can't thread ban a word.

This sounds like wanting to ban the word "Euclidean" in a discussion of hyperbolic geometry and the fifth postulate

 

[PeroK]

When one takes the same quantum state, but one changes the observation basis, we get a DIFFERENT probability distribution over ANOTHER universe of possible outcomes. These are ALSO standard probabilities, but as we have now a totally different set of outcomes, there's no link with the previous universe.

The probabilities and correlations are perfectly viable. But not with the physical assumptions in Bell's theorem. The simplest way to see that they are viable is to imagine a correlation between the state to be measured and the measuring device.

Moreover, a state is not a probability distribution. It involves complex probability amplitudes. It's in that underlying structure where QM explicity and fundamentally diverges from the classical theories. Amplitudes can cancel, whereas real probabilities can only add. That has nothing to do with entanglement specifically.

Finally, in all cases (including entanglement) the probabilities predicted by QM satisfy the Kolmogorov axioms. They are real probabilities, in other words.

 

[Fra]

TL;DR Summary: I don't understand entanglement

I don't seem to wrap my head around what really happens to an entangled system during a local measurement.

Does anyone understand this?

Most are happy to just beeing able to describe the correlation of outcomes. The local mechanisms, are still an open question I would say, and it's what feeds many threads on here.

/Fredrik

 

[Sargon38]

The probabilities and correlations are perfectly viable. But not with the physical assumptions in Bell's theorem. The simplest way to see that they are viable is to imagine a correlation between the state to be measured and the measuring device.

Is that so ?
My idea was that there doesn't exist a global probability universe that *at the same time* can explain the different angular correlations in a typical Bell experiment. Of course, one can sneak out by saying that there are "non-local interactions as a result of the changing measurement" but that exactly comes down to saying that there was no a priori outcome distribution. Which is the essence of what I'm saying: there is no a priori outcome distribution over the counterfactual outcomes. That's what entanglement is about.

That is, if you have an entangled pair of the singlet state 2 spin-1/2 particles, the quantum mechanical prediction of the correlation between clicks if measured "up" on both sides is:
$P(\theta_1, \theta_2) = \frac {1}{2} \sin^2 \left( \frac{\theta_1 - \theta_2}{2} \right)$
Now, the whole point is that there doesn't exist a Kolmogorov probability distribution over the universe of all possible events $\Omega = \{ (\theta_1,\theta_2,click/no-click,click/no-click) | \theta_1,\theta_2 \in [0,\pi/2] \}$ that will reproduce the above correlations.

The reason is that if such a distribution existed, then there is an inequality to be satisfied:
P(A and not B) + P(B and not C) > P(A and not C) which can easily be verified on a Venn diagram with 3 intersecting sets A, B and C (and we use Kolmogorov's axioms).
Indeed, the first two sets are disjoint, so the sum of the probabilities corresponds to the probability of the union, and (A and not B) union (B and not C) includes all of (A and not C), plus extra pieces.

Well, this isn't the case for the correlations between theta, if we pick 0, 45 and 90 degrees and we look at
P(0,45,click,no-click), P(45,90,click,no-click) and P(0,90,click,no-click).
Namely, $1/2 \sin^2 22.5$ for the first two, which amount to 0.073... and the last one which is 0.25.

And 0.073 + 0.073 is not bigger than 0.25, what was supposed to be the case if all these correlations found their origin in a single master probability universe.

This amounts to saying that there doesn't exist a global probability distribution of all (counterfactual) outcomes (at least, a Kolmogorov probability distribution).

I know what you mean by "physical assumptions" but they come down to "not being able to propagate the choice of the measurement apparatus on side 1 to side 2". However, that is already implicit in the idea that there was an a priori probability distribution over all counterfactual outcomes.

Moreover, a state is not a probability distribution. It involves complex probability amplitudes. It's in that underlying structure where QM explicity and fundamentally diverges from the classical theories. Amplitudes can cancel, whereas real probabilities can only add. That has nothing to do with entanglement specifically.

Finally, in all cases (including entanglement) the probabilities predicted by QM satisfy the Kolmogorov axioms. They are real probabilities, in other words.

I fully agree that a state is not a probability distribution. That's exactly the point in fact. If a state WERE a probability distribution, we wouldn't have these violations over counterfactual outcomes.

A quantum state + a measurement basis provides us with a probability distribution. For a same state, but a different basis, we get a different probability distribution, over a different set of outcomes.

 

[Sargon38]

I fully agree that a state is not a probability distribution. That's exactly the point in fact. If a state WERE a probability distribution, we wouldn't have these violations over counterfactual outcomes.

A quantum state + a measurement basis provides us with a probability distribution. For a same state, but a different basis, we get a different probability distribution, over a different set of outcomes.

To continue with this, I wonder in fact if we can have a similar demonstration of violation of existence of counterfactual probabilities on a "simple" quantum state that one wouldn't call "an entangled state", say a single scalar particle system.

Is there a way to have different measurements on a single scalar particle state, that violate evident probability properties if they were taken counterfactually to exist together ?

My idea was that we can't, but maybe there are examples of such.

Of course, one is tempted to say "the double slit experiment" where there's an obvious clash between the probability of going through slit 1, going through slit 2 and hitting the screen at position x, but the problem is that this isn't the same state when looking at the holes, and looking at the screen, there's an evolution between when the particle goes through the slits, and hits the screen.

 

[PeroK]

My idea was that there doesn't exist a global probability universe that *at the same time* can explain the different angular correlations in a typical Bell experiment. Of course, one can sneak out by saying that there are "non-local interactions as a result of the changing measurement" but that exactly comes down to saying that there was no a priori outcome distribution. Which is the essence of what I'm saying: there is no a priori outcome distribution over the counterfactual outcomes. That's what entanglement is about.

That's not what entanglement is about. It's not about breaking classical probability theory. It's about breaking the physical assumptions in the Bell theorem. You can model the outcomes of a Bell experiment using classical probability theory by defining the sample space not in terms of individual particles (with their specific probabilities), but in terms of all the overall experiment, where all the correlations are built in. In other words, the sample space includes the factor of what is being measured.

There is a analogy in what you are doing with the argument that the relativistic velocity addition breaks the rules of arithmetic, because $\frac c 2 + \frac c 2 \ne c$. That's not what breaks. What breaks is the assumption that a velocity transformation in that case is a simple addition. You are doing something more subtle and sophisticated with probability theory and QM. But, it has the same underlying flaw.

 

[Sargon38]

That's not what entanglement is about. It's not about breaking classical probability theory. It's about breaking the physical assumptions in the Bell theorem. You can model the outcomes of a Bell experiment using classical probability theory by defining the sample space not in terms of individual particles (with their specific probabilities), but in terms of all the overall experiment, where all the correlations are built in. In other words, the sample space includes the factor of what is being measured.

That's trivially true because of course ANY experiment that can really be done will ALWAYS be describable by standard probabilities. Nothing that can really be done in the lab will ever "break" standard probability axioms. There is of course no "executable set of observations" which can ever violate standard probabilities that would describe their outcomes.

My point was that what entanglement is about, is to be incompatible with a statistical mixture. If that weren't the case, there wouldn't be ANY reason to work in product Hilbert spaces for the quantum states of compound systems (which is what entanglement is about: the fact that we work in a direct product of Hilbert spaces of the sub systems), because ANY correlation between subsystems would be describable by statistical mixtures. So that was to me what characterises entanglement: that it CANNOT be described by a statistical mixture of any nature.

But of course, to even say what is meant by a statistical mixture, you need to assume that the different measurements you're going to perform are going to measure in the same "probability universe". So that the probability universe is not ALTERED by the particular measurement you're going to perform.
This is what you call the "physical assumptions" behind Bell's derivation, namely that by changing the measurement you do on particle 1, you're not going to ALTER the probability distribution that was supposed to describe them all. In other words, the assumption of a statistical mixture is of course that the probability distribution over the entire universe is not going to be altered by the aspect you want to measure.

In other words, there's a fundamental difference between $|up> \otimes |up> - | down > \otimes |down>$ and 50% of $|up> \otimes |up>$ and 50% of $| down > \otimes |down>$. That's what entanglement is about, that there's NO WAY to find a statistical mixture that would give, an all circumstances, the same outcome probabilities.

 

[Sargon38]

There is a analogy in what you are doing with the argument that the relativistic velocity addition breaks the rules of arithmetic, because $\frac c 2 + \frac c 2 \ne c$. That's not what breaks. What breaks is the assumption that a velocity transformation in that case is a simple addition. You are doing something more subtle and sophisticated with probability theory and QM. But, it has the same underlying flaw.

That is in fact a very good analogy, as far as analogies go.

What I'm saying here is indeed equivalent to: "what characterises the relativistic composition of velocities, is that you can't use the vector addition of velocities".
If people would ask me what is the characteristic of relativistic effects, I would say exactly that: vector addition of velocities in different frames doesn't work any more.

So if people ask what's characteristic of entanglement, I say: the fact that you cannot derive the correlations of the measurements of subsystems from a single overall standard probability distribution that would describe them all. That you can't use a statistical mixture to describe all the correlations that are possible.

 

[PeterDonis]

there are equivalent terms like "standard" probabilities, "classical" probabilities and so on because I had the impression people got nervous at the term Kolmogorov.

So by all of those terms you just mean "Kolmogorov"? If so, just use one term. We don't get nervous here.

which is why "Kolmogorov" seemed more to the point to me, but this had some enerving effect on Dr Chinese

That's not because he doesn't understand what "Kolmogorov probabilities" means. It's because he doesn't agree with you on its relevance to the current discussion. Making up new terminology for a known concept doesn't help with that at all.

And "standard" might also be mis-interpreted as being something else than "the probabilities satisfying the standard properties of a probability distribution over a universe".

Now I'm confused again. By "standard" do you mean "Kolmogorov" or something else?

I try once more to reformulate what I was saying in this thread: the *characteristic* of entangled states is that they imply "probability distributions" (and specifically under the form of correlations) which are NOT Kolmogorov under non-contextual hypotheses (usually called hidden variable hypotheses). I put "probability distributions" in quotes because their Kolmogorov-violating aspects make that it aren't probability distributions.

This is to me what is the essence of entanglement, and what distinguishes it fundamentally from a statistical mixture, where, of course, by definition, the generated probability distribution (and hence the correlations that are part of it) would satisfy the Kolmogorov axioms.

All of this can be stated much more simply: entangled states can produce correlations that violate the Bell inequalities (or their equivalents for a particular scenario). Yes, we all know that. And adding lots of talk about "Kolmogorov" and "violating Kolmogorov" adds nothing to our knowledge. It just restates what we already know in much more opaque language.