How Does Local Measurement Affect an Entangled System?

[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.

 

[Sargon38]

We don't get nervous here.

That's not my impression. Very long ago when I was a mentor here under my real world name, these kinds of discussions where less agressive and more enlightening.

 

[Fra]

My take with entanglement is an MWI like view.

I saw this first now, interpreting is easiser if you see the stance. I now understand why refer to universes below.

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.

In contrast, my take on entanglement fits in an "interacting agent/observer" interpretation.

Here the explanation of the correlation is simply a "Reichenbach's Common cause".
+
The explanation of why Bell ansatz is wrong (and thus we don't expect it to hold in the first place) is that interactions is understood in a way there each subsystem of the universe, reacts towards it's environment, not based in hidden variables, but randomly based on what it knows. And in the case of the bell experiment, as the entangled pais is explicitly isolated after creation, not only to the phycisists but from ANY inteaction, when the "instruments know" is only the information implicit in the "preparation". Thus to expect that the instrument would interact as if the hidden variable was known, would not be rational.

This is why the bell inequality to me at least is not a mystery per see, the open question for me is to understand find how the hamiltonian och nature and the fundamental interactions emerge in line with this interpretation.

Similarly the explanation for the complex state in this same interpretation is simply that the agent needs a may to merge information coming from several DEPENDENT sampling spaces. For example the simplest case is trying to merge into one "state" information about position and also momentum, and if you define conjugate momenta via fourier transforms, the complex state is the natural solution in compact notation. But it would be possible to consider instead a combination of difference spaces, that are related. But there are still many open issues in trying to make this complete and consistent, and until that is int place I think it's not really understood.

/Fredrik

 

[Morbert]

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

I don't think I understand this claim. By probability set, are you referring to a probability measure? What does it mean to have a sample space for contrafactual measurements?

Given a classical system and classical theory, the sample spaces of two different measurement procedures are always coarse-grainings of a third sample space. This is not true for a quantum system and theory. E.g. The sample space for a measurement of spin-x of a particle, and the sample space for measurement of spin-y of a particle do not imply a third sample space for measuring "spin-x and spin-y" of a particle. Is this what you mean? This is true whether or not there is entanglement.

 

[PeroK]

This is true whether or not there is entanglement.

For example, the double-slit pattern (which is effectively a probability distribution) is not the classical average of the probability distributions of the two single slits. And, again, its classical physical assumptions that fail, rather than classical probability theory itself.

 

[PeterDonis]

Moderator's note: Moved thread to interpretations subforum.