Statistical Independence in Quantum Mechanics

[Lynch101]

TL;DR Summary

Baisc question about statistical independence in QM - are they simple correlations or is there a nuance?

Very basic question here, about statistical independence in quantum mechanical experiments. The quote from PD below is what prompted the question.

We are talking about a lack of statistical independence between a photon source at A and light sources in two quasars, each a billion light-years from A in opposite directions. How else could that possibly be except by some kind of pre-existing correlation?

When we talk about "some kind of pre-existing correlation" are talking about a simple correlation in the sense of the correlation of sunglasses and ice creams, or is a more nuanced type of correlation (of which I am unfamiliar) meant?

I understand the issue of the photons from the quasars having been emitted a billion years earlier, but is there any other nuance that is at play?

 

[PeterDonis]

When we talk about "some kind of pre-existing correlation" are talking about a simple correlation in the sense of the correlation of sunglasses and ice creams, or is a more nuanced type of correlation (of which I am unfamiliar) meant?

The thread you quoted from was about superdeterminism, so the "pre-existing correlation" in question is the kind that superdeterminism requires in order to account for the kind of experimental results that were being described.

 

[Lynch101]

The thread you quoted from was about superdeterminism, so the "pre-existing correlation" in question is the kind that superdeterminism requires in order to account for the kind of experimental results that were being described.

In the context of statistical independence (leaving superdeterminism aside) is it a question of a simple correlation as in the sunglasses and the ice cream or is there some nuance to it?

Presumably there is some correlation between the light source from from the quasars and the photon source for the experiment but, presumably, that isn't sufficient to violate statistical independence, is it?

 

[PeterDonis]

simple correlation as in the sunglasses and the ice cream

I have no idea what you mean by this.

Presumably there is some correlation between the light source from from the quasars and the photon source for the experiment

Why would we presume this? The photon source is here on Earth. The quasars are billions of light years away. Why would there be any correlation between them?

 

[Lynch101]

I have no idea what you mean by this.

When the sun is shining, there may be a correlation between the number of people eating ice creams and the number of people wearing sunglasses.

Why would we presume this? The photon source is here on Earth. The quasars are billions of light years away. Why would there be any correlation between them?

Are they not correlated by virtue of being present in the universe at the same time? As in, the photon used to choose the detector settings is present in the universe at the same moment as the photon is emitted on earth. That is a correlation, isn't it?

If so, is it sufficient to violate statistical independence?

 

[PeterDonis]

When the sun is shining, there may be a correlation between the number of people eating ice creams and the number of people wearing sunglasses.

Ok, and since the Sun is obviously in the common past light cone of the ice creams and the sunglasses, it can serve just fine as a common cause of both. And it should be obvious that, since a quasar a billion light years away is not in the common past light cone of all of the pieces of an experimental setup such as I described, this setup is not the same as your ice creams/sunglasses setup.

I think this is a case (not the first with you) of you not thinking very carefully about what you are asking before you ask it.

 

[PeterDonis]

Are they not correlated by virtue of being present in the universe at the same time?

Why would they be? Why would you expect a quasar a billion light years away to be "correlated" with anything here on Earth?

the photon used to choose the detector settings is present in the universe at the same moment as the photon is emitted on earth.

"At the same moment" makes no sense. The event at which the photon used to choose the detector settings is emitted is spacelike separated from the event at which the photon to be measured by the detector is emitted on Earth. That is the only invariant statement that can be made. There is no invariant "moment" that those events have in common.

 

[Lynch101]

Why would they be? Why would you expect a quasar a billion light years away to be "correlated" with anything here on Earth?

Does it have to be the source (quasars) that are correlated or just the photons from the quasars?

It's not that I expect it, I'm just wondering about it. If we think in terms of a rocket that left from a distant galaxy, let's say a billion light years away and it zooms past Earth when a baseball pitcher is throwing a pitch, the rocket and the pitch would be correlated, in some reference frame, wouldn't they?

"At the same moment" makes no sense. The event at which the photon used to choose the detector settings is emitted is spacelike separated from the event at which the photon to be measured by the detector is emitted on Earth. That is the only invariant statement that can be made. There is no invariant "moment" that those events have in common.

Does that mean that such a correlation that would violate statistical independence is impossible, or is there a way in which correlations make sense without referring to being present "at the same moment"?

 

[PeterDonis]

Does it have to be the source (quasars) that are correlated or just the photons from the quasars?

Since the photons' states are determined at the source, it's the same thing.

If we think in terms of a rocket that left from a distant galaxy, let's say a billion light years away and it zooms past Earth when a baseball pitcher is throwing a pitch, the rocket and the pitch would be correlated, in some reference frame, wouldn't they?

Why would they?

Does that mean that such a correlation that would violate statistical independence is impossible, or is there a way in which correlations make sense without referring to being present "at the same moment"?

I have no idea. You would have to ask a superdeterminist.

You seem to have this backwards. The default assumption for objects that come from spacelike separated sources is not that they are correlated, it's that they aren't correlated. That's why so many people object to superdeterminism: because it has to claim that, despite everyone's natural expectation to the contrary, such objects are correlated, in just the right way to make experimental results match the predictions of QM.

 

[gentzen]

It's not that I expect it, I'm just wondering about it. If we think in terms of a rocket that left from a distant galaxy, let's say a billion light years away and it zooms past Earth when a baseball pitcher is throwing a pitch, the rocket and the pitch would be correlated, in some reference frame, wouldn't they?

Correlation is a statistical property, namely X and Y are uncorrelated if E[XY]=E[X] E[Y], i.e. if their correlation coefficient is zero. So you need a reasonable expectation value functional E[.] to be defined, before it even makes sense to talk of correlation. For a one-shot event like a rocket zooming past earth, there is no reasonable expectation value functional E[.] that would allow to make sense of your question.

The reason for talking about correlations in the first place is that they are so well defined and measurable.

You wonder that statistical independence is similarly well defined in that X and Y are statistical independent iif E[f(X)g(Y)]=E[f(X)] E[g(Y)] for arbitrary functions f and g. However, that is not the case, because a reasonable expectation value functional E[.] comes with some variance (uncertainty) that allows to make "robust sense" of the equation E[XY]=E[X] E[Y]. But there is nothing similar that would allow to make "robust sense" of the equation E[f(X)g(Y)]=E[f(X)] E[g(Y)].

 

[Lynch101]

Since the photons' states are determined at the source, it's the same thing.

I see thank you.

Why would they?

That's just what I thought correlation was, where two things appear together. Another example I had in mind was the appearance of rain clouds in the sky and the number of people carrying umbrellas. That is an example of a correlation isn't it?

I have no idea. You would have to ask a superdeterminist.

You seem to have this backwards. The default assumption for objects that come from spacelike separated sources is not that they are correlated, it's that they aren't correlated. That's why so many people object to superdeterminism: because it has to claim that, despite everyone's natural expectation to the contrary, such objects are correlated, in just the right way to make experimental results match the predictions of QM.

I think my misunderstanding lies in what constitutes a correlation. I was thinking that the examples I gave were examples of correlations and that the same would apply to quantum experiments.

 

[Lynch101]

Correlation is a statistical property, namely X and Y are uncorrelated if E[XY]=E[X] E[Y], i.e. if their correlation coefficient is zero. So you need a reasonable expectation value functional E[.] to be defined, before it even makes sense to talk of correlation. For a one-shot event like a rocket zooming past earth, there is no reasonable expectation value functional E[.] that would allow to make sense of your question.

The reason for talking about correlations in the first place is that they are so well defined and measurable.

You wonder that statistical independence is similarly well defined in that X and Y are statistical independent iif E[f(X)g(Y)]=E[f(X)] E[g(Y)] for arbitrary functions f and g. However, that is not the case, because a reasonable expectation value functional E[.] comes with some variance (uncertainty) that allows to make "robust sense" of the equation E[XY]=E[X] E[Y]. But there is nothing similar that would allow to make "robust sense" of the equation E[f(X)g(Y)]=E[f(X)] E[g(Y)].

Thabks Gentzen, I'm not familiar with this explanation with regard to correlation. I've only ever encountered explanations of correlations with regard to empirical observations; as in the examples above e.g. the observation of rain clouds in the sky and the observation of people carrying umbrellas.

Is expectation value above similar to a Bayesian probability function or Bayesian inference?

 

[gentzen]

Is expectation value above similar to a Bayesian probability function or Bayesian inference?

Take whatever reasonable expectation value functional that is available in your given context. Normally this just means to take the average of your observations, i.e. the empirical mean (or sample mean).

 

[Lynch101]

Thanks Gentzen.

Take whatever reasonable expectation value functional that is available in your given context.

Does this mean calculate the expectation value in the context of the given theory/interpretation?

Normally this just means to take the average of your observations, i.e. the empirical mean (or sample mean).

Would this just be the mean of the observations in quantum experiments?

 

[gentzen]

Does this mean calculate the expectation value in the context of the given theory/interpretation?

That is certainly allowed. But it is not the only option.

Would this just be the mean of the observations in quantum experiments?

Yes, that is certainly also allowed. However, there is the caveat for hidden variable theories (and hence superdeterminism) that you cannot observe the hidden variables, and hence you cannot determine their empirical mean (sample mean).

 

[Lynch101]

That is certainly allowed. But it is not the only option.


Yes, that is certainly also allowed. However, there is the caveat for hidden variable theories (and hence superdeterminism) that you cannot observe the hidden variables, and hence you cannot determine their empirical mean (sample mean).

Is it possible for one off events to be correlated, such that there is no meaningful average of observations (or the average is the value associated with the single observable - if that makes sense)?

 

[gentzen]

Is it possible for one off events to be correlated

No, you need to use a different word in that case, which expresses what you want to say. Probably you want something along the lines that both events have a common cause.

 

[Lynch101]

No, you need to use a different word in that case, which expresses what you want to say. Probably you want something along the lines that both events have a common cause.

Thank you.

 

[Jarvis323]

You might think there should be some correlation which comes from both events involving and stemming from phenomena in the same universe with presumably the same physical constants and where the same laws of physics apply. And there is some correlations of that nature that exist between phenomena there and here, even though the two places are too far apart to be recently causally linked.

But we can still define a measurement of the event or phenomena and with what we know about the common physics, we can assume statistical independence on the basis of theory which aggrees with observation here. Ironically, it's our assumption that the two places are correlated that we are able to say that the specific events we measure should be uncorrelated.

Also, the universe is full of chaos, which, in theory, deterministically makes some things look randomized, and is able to make correlations between events far too small and impossible for us to deduce even when events are causally linked. That chaos is a deterministic process means superdeterminism could still somehow possibly be facilitated by chaotic processes, but for that to be possible you would at least need some really precise initial conditions or previous conditions which are hard to imagine somehow are always just right to give us these unexpected correlations.

Since chaos randomizes things, Sabine H. who is a top proponent if SD, says to look for evidence of the correlations we should look for correlations between events at small scales which are close in time and space, before chaos takes over and makes it impossible to detangle. Note that Sabine doesn't define superdeterminism in a way that necessarilly requires correlation on the origin of the two events which are separated so much. By the time an event there has a measurable effect here, it has interacted with things here. So the correlations, she would say, can come from those interactions. She only requires the correlations are somehow formed at some point in the past. Note that it is only through the interaction with detectors that the measurements between the two events become known to us. And ultimately it is the measurements (both taken here on Earth) that we are comparing. She advocates thus to focus on the interactions with detectors.

The problem is there is no theoretical reason why any such interactions should create those correlations. So without observations that demonstrate they do, it is just a speculation and many estimate it to be implausible.

Basically correlation doesn't imply causality but some correlations are hard to imagine can be feasible without it. And causality doesn't imply measurable correlation, but if we look at small enough scales in
time and space, deterministic causality should imply correlation (assuming we can look close enough and the theory is reliable).

 

[Jarvis323]

Is it possible for one off events to be correlated, such that there is no meaningful average of observations (or the average is the value associated with the single observable - if that makes sense)?

Technically every event is one off in its full specification. We reduce the specification to the meaningful aspects which relate them with our theory and questions about them. Then we ask about correlations about the classes of similar events.

 

[PeterDonis]

That's just what I thought correlation was, where two things appear together.

This is much too simplistic. See below.

Another example I had in mind was the appearance of rain clouds in the sky and the number of people carrying umbrellas. That is an example of a correlation isn't it?

Not a single instance of it, no. We say there is a correlation between rain clouds and people carrying umbrellas if the probability of people carrying umbrellas is higher when there are rain clouds in the sky than when there aren't. Or vice versa. But you can't calculate a probability from a single event. You have to have a large number of events that fall into some relevant category.

 

[Lynch101]

This is much too simplistic. See below.


Not a single instance of it, no. We say there is a correlation between rain clouds and people carrying umbrellas if the probability of people carrying umbrellas is higher when there are rain clouds in the sky than when there aren't. Or vice versa. But you can't calculate a probability from a single event. You have to have a large number of events that fall into some relevant category.

Thanks Peter.

 

[Lynch101]

You might think there should be some correlation which comes from both events involving and stemming from phenomena in the same universe with presumably the same physical constants and where the same laws of physics apply. And there is some correlations of that nature that exist between phenomena there and here, even though the two places are too far apart to be recently causally linked.

But we can still define a measurement of the event or phenomena and with what we know about the common physics, we can assume statistical independence on the basis of theory which aggrees with observation here. Ironically, it's our assumption that the two places are correlated that we are able to say that the specific events we measure should be uncorrelated.

Also, the universe is full of chaos, which, in theory, deterministically makes some things look randomized, and is able to make correlations between events far too small and impossible for us to deduce even when events are causally linked. That chaos is a deterministic process means superdeterminism could still somehow possibly be facilitated by chaotic processes, but for that to be possible you would at least need some really precise initial conditions or previous conditions which are hard to imagine somehow are always just right to give us these unexpected correlations.

Since chaos randomizes things, Sabine H. who is a top proponent if SD, says to look for evidence of the correlations we should look for correlations between events at small scales which are close in time and space, before chaos takes over and makes it impossible to detangle. Note that Sabine doesn't define superdeterminism in a way that necessarilly requires correlation on the origin of the two events which are separated so much. By the time an event there has a measurable effect here, it has interacted with things here. So the correlations, she would say, can come from those interactions. She only requires the correlations are somehow formed at some point in the past. Note that it is only through the interaction with detectors that the measurements between the two events become known to us. And ultimately it is the measurements (both taken here on Earth) that we are comparing. She advocates thus to focus on the interactions with detectors.

The problem is there is no theoretical reason why any such interactions should create those correlations. So without observations that demonstrate they do, it is just a speculation and many estimate it to be implausible.

Basically correlation doesn't imply causality but some correlations are hard to imagine can be feasible without it. And causality doesn't imply measurable correlation, but if we look at small enough scales in
time and space, deterministic causality should imply correlation (assuming we can look close enough and the theory is reliable).

Thanks Jarvis.