The Efficiency Loophole: A Local Hidden Variables Theory?

[JesseM]

Thanks for the summary, but those assertions are a little (too) extreme.
I would think that it is the assertion of the Local Realist that there is no magical dependence of a measurement here on a result there (separability/locality). Influences at a distance according to known or not yet known physical mechanisms are admitted. However I agree that that does essentially deny physical entanglement at a great distance. [..] Further, a Local Realist assumes that already before the measurement one or more unobserved particle variables exist that will affect the values that will be measured. And I suppose that Bell's theorem is meant to apply to such local realism.

The meaning of "locality" specifically has to do with there being no FTL causal influences...if there are such FTL influences this would be a violation of locality, it's irrelevant whether the influences obey some well defined "physical mechanism" or if they appear "magical" to us.

 

[DrChinese]

"Perfect" correlations don't exist in scientific measurements - scientific measurements work with measurement errors.

You got me there!

 

[harrylin]

The meaning of "locality" specifically has to do with there being no FTL causal influences...if there are such FTL influences this would be a violation of locality, it's irrelevant whether the influences obey some well defined "physical mechanism" or if they appear "magical" to us.

FTL would be a violation of some interpretations of SR. Quite some people accept the possibility of FTL influences as long as they cannot be used for signaling. A "FTL influence" even implies the concept of locality: with a non-local concept there is no locality to influence another one.
However, probably this is not really relevant for loopholes.

 

[JesseM]

FTL would be a violation of some interpretations of SR. Quite some people accept the possibility of FTL influences as long as they cannot be used for signaling. A "FTL influence" even implies the concept of locality: with a non-local concept there is no locality to influence another one.
However, probably this is not really relevant for loopholes.

I don't think you're using "locality" the way most physicists use it, there is no possibility of signaling in the Bohmian interpretation but no one would call this a "local" interpretation, and "locality" does not just mean that causal influences are transmitted by discrete localizable particles which might nevertheless be moving FTL as you seem to imply, it specifically is used to refer to the idea that no causal influences move FTL.

 

[harrylin]

I don't think you're using "locality" the way most physicists use it, there is no possibility of signaling in the Bohmian interpretation but no one would call this a "local" interpretation, and "locality" does not just mean that causal influences are transmitted by discrete localizable particles which might nevertheless be moving FTL as you seem to imply, it specifically is used to refer to the idea that no causal influences move FTL.

I kind of agree with that (I see that I didn't formulate it well) but we drifted far away from "local realist" on which we probably already agreed (I wrote: "I agree that that does essentially deny physical entanglement at a great distance"). The meaning of such words as "locality" depend on the context, and here I take it to refer to the physical process of detection which a Local Realist supposes to be of negligible influence far away - thus a "local" process. I think that "FTL" isn't the essential point of "locality" - nor is this little excursion relevant for loopholes.

 

[JesseM]

We drifted far away from "local realist" on which we probably already agreed ("I agree that that does essentially deny physical entanglement at a great distance"). The meaning of such words as "locality" depend on the context

In the specific context of Bell and local realism, I think "locality" is generally used to forbid FTL causal influences. It might be that in some other contexts "locality" simply means that no causal influences are infinitely fast (instantaneous), but this definition would be completely untestable since no experiment can ensure that the time between the two distant measurements is precisely zero. Can you find any examples of physicists using "locality" in a way that allows FTL "local" influences in the context of Bell/local realism?

here I take it to refer to the physical process of detection which a Local Realist supposes to be of negligible influence far away - thus a "local" process.

I don't know what "of negligible influence far away" means, if my detector's interaction with the particle causes a localized FTL influence to be transmitted to the other particle which influences how that particle interacts with the other detector, isn't that a non-negligible influence?

 

[harrylin]

In the specific context of Bell and local realism, I think "locality" is generally used to forbid FTL causal influences. It might be that in some other contexts "locality" simply means that no causal influences are infinitely fast (instantaneous), but this definition would be completely untestable since no experiment can ensure that the time between the two distant measurements is precisely zero. Can you find any examples of physicists using "locality" in a way that allows FTL "local" influences in the context of Bell/local realism?

No, although at one point I poorly formulated it, I stressed that in the context of Bell/local realism ("Bell non-locality"), "locality" is used to indicate a local process or interaction* - thus without measurable effect at a great distance. I'm simply against jargon creep.

* "requiring the value assigned to an operator associated with an individual constitutent to be independent of what is measured on any other constitutent"
- http://plato.stanford.edu/entries/bell-theorem/

I don't know what "of negligible influence far away" means, if my detector's interaction with the particle causes a localized FTL influence to be transmitted to the other particle which influences how that particle interacts with the other detector, isn't that a non-negligible influence?

Certainly that is a non-negligible and non-local process. Again: "locality" in this context is about the influence of detection which is supposed to be limited to the environment of the detection, not about FTL. As I understand it, "Bell non-locality" is the negation of that classical assumption.

Harald

 

[JesseM]

No, although at one point I poorly formulated it, I stressed that in the context of Bell/local realism ("Bell non-locality"), "locality" is used to indicate a local process or interaction* - thus without measurable effect at a great distance. I'm simply against jargon creep.

That seems too vague for a technical term though--what are the precise limits of "great distance"? And surely if enough time has passed then effects at great distances are OK (for example a message sent from Earth to Alpha Centauri today may have a measurable effect there in 4.4 years!), it's only quasi-instantaneous effects at great distances that are a problem?

Also, it occurs to me that if you accept the relativity of simultaneity, then there is really no meaningful distinction between defining locality as "no instantaneous influences" and "no FTL influence"--after all, a particle that travels from location A to location B at a speed even slightly faster than light in one frame will depart A at precisely the same time it arrives at B in some other frame.

 

[harrylin]

That seems too vague for a technical term though--what are the precise limits of "great distance"? And surely if enough time has passed then effects at great distances are OK (for example a message sent from Earth to Alpha Centauri today may have a measurable effect there in 4.4 years!), it's only quasi-instantaneous effects at great distances that are a problem?

Also, it occurs to me that if you accept the relativity of simultaneity, then there is really no meaningful distinction between defining locality as "no instantaneous influences" and "no FTL influence"--after all, a particle that travels from location A to location B at a speed even slightly faster than light in one frame will depart A at precisely the same time it arrives at B in some other frame.

OK, "great" of "great distance" is more for a convincing experiment. But sorry, I can't change the meaning of "local"! One last time: "local" has no direct relationship with speed of transmission. In the context of Bell it simply means that a measurement at one place does not affect the measurement outcome at another place. And the more clearly the particles are separated, the more clearly they are at different locations.

As formulated by Shimony: "Locality is a condition on composite systems with spatially separated constituents, requiring [..] the value assigned to an operator associated with an individual constitutent to be independent of what is measured on any other constitutent."
- http://plato.stanford.edu/entries/bell-theorem/

Harald

PS I suppose that we all agree that the term "local hidden variable theory" points to a theory in which measurement related variables exist before they are measured, and whereby the measurement on one of two spatially separated entities does not affect the measurement outcome on the other entity.

 

[JesseM]

OK, "great" of "great distance" is more for a convincing experiment. But sorry, I can't change the meaning of "local"! One last time: "local" has no direct relationship with speed of transmission. In the context of Bell it simply means that a measurement at one place does not affect the measurement outcome at another place.

But that would suggest that it's irrelevant to Bell's proof whether or not there is a spacelike separation between the two measurements or a timelike one, as long as there is a "great distance" between the two measurements. Do you really think that's the case? What about the locality loophole in Bell experiments?

Also, in Bell's own paper La nouvelle cuisine, much of which can be read on google books starting on p. 216 of this book (it's also available in Speakable and Unspeakable in Quantum Mechanics), he specifically defines his notion of "local causality" in terms of the speed of light limit, and this plays an essential role in his derivation. On p. 217 he starts out connecting locality and the speed of light:

I will be particularly concerned with the idea that effects are near to their causes:

"If the results of experiments on free fall here in Amsterdam would depend appreciably on the temperature of Mont Blanc, on the height of the Seine below Paris and on the position of the planets, one would not get very far.", H.B.G. Casimir.

Now at some very high level of accuracy, all these things would become relevant for free fall in Amsterdam. However even then we would expect their influence to be retarded by at least the time that would be required for the propagation of light. I will be much concerned here with the idea of the velocity of light as a limit.

Then on p. 224 he gives a definition of his "principle of local causality", saying:

The direct causes (and effects) of events are near by, and even indirect causes (and effects) are no further away than permitted by the velocity of light.

Then on p. 225 he makes this more precise with a diagram of the past light cones of two regions 1 and 2, with region 3 being a complete cross-section of the past light cone of region 1 (bounded above and below by spacelike surfaces) which is "above" the region where the two past light cones overlap, so that no point in 3 is part of the overlap region, and any timelike or lightlike worldline which starts from an event in the past light cone of 2 would have to pass through region 3 in order to pass through region 1. Then he defines "local causality" more precisely in terms of this diagram (and also in terms of "local beables", local physical facts taken to be basic elements of any local theory of physics, which he had earlier brought up on p. 219), saying:

A theory will be said to be locally causal if the probabilities attached to the values of local beables in a space-time region 1 are unaltered by specification of values of local beables in a space-like separated region 2, when what happens in the backward light cone of 1 is already sufficiently specified, for example by a full specification of all local beables in a space-time region 3 (figure 6.4).

So you can see that the notion that local facts in region 1 don't depend on facts at a space-like separation in region 2 (and that any correlation can be "screened off" by including facts about another region 3 that any causal influences from the overlap of the two post light cones would have to travel through to get to 1) plays a critical role in his argument, and is the justification for the step on p. 228 where he starts with equation 6.9.2, {A,B|a,b,c,λ}={A|B,a,b,c,λ}{B|a,b,c,λ} (here a and b refer to detector settings in region 1 and 2, while c refers to observable variables in region 3 and λ refers to hidden variables in region 3) and then uses local causality to get the next equation:

Invoking local causality, and the assumed completeness of c and λ in the relevant parts of region 3, we declare redundant certain of the conditional variables in the last expression, because they are at space-like separation from the result in question. Then we have

{A,B|a,b,c,λ}={A|a,c,λ}{B|b,c,λ}

And this step is essential to his proof that QM cannot be explained by a "locally causal" theory of hidden variables.

 

[harrylin]

But that would suggest that it's irrelevant to Bell's proof whether or not there is a spacelike separation between the two measurements or a timelike one, as long as there is a "great distance" between the two measurements. Do you really think that's the case? What about the locality loophole in Bell experiments?

Not at all: for testing theories against each other it is essential to test situations where there can be no doubt that they predict something different; if one fails to do so, one creates a loophole*. For those (like Einstein) who accept Relativity, local causality implies that no influence towards another location can occur faster than light.

However, you do make a good argument here below that Bell defined local causality slightly different from what I am used to; see next.

Also, in Bell's own paper La nouvelle cuisine, much of which can be read on google books starting on p. 216 of this book (it's also available in Speakable and Unspeakable in Quantum Mechanics), he specifically defines his notion of "local causality" in terms of the speed of light limit, and this plays an essential role in his derivation. On p. 217 he starts out connecting locality and the speed of light:

Thanks for the very useful link! There he starts out with the common meaning of local causality:

"the idea that effects are near to their causes",

and next indeed he states that he will be very much concerned with the limit of the speed of light.

Then on p. 224 he gives a definition of his "principle of local causality", saying:

"The direct causes (and effects) of events are near by, and even indirect causes (and effects) are no further away than permitted by the velocity of light."

Indeed. Note that he doesn't suggest that a theory would be necessary "non-local" if events happen close enough to each other to be within the light cone. And what you did not cite:

"Here we have preferred to see it not as a formulation of local causality but as a consequence thereof."
("It" = factorizability due to independence of A on B and vice versa).

[..] And this step is essential to his proof that QM cannot be explained by a "locally causal" theory of hidden variables.

Yes indeed - thanks for your nice summary.

I find Shimony's definition natural and linguistically pure, while I now see that indeed Bell's is subtly different; thanks for pointing that out. Note that it has no consequence for the derivation. Bell admitted that what is "very often" done (differently form his formulation), is to define that A and B do not depend on each other nor on remote polarizers. (p.228/109).Harald

*PS: Shimony calls it the "communication loophole"
- http://plato.stanford.edu/entries/bell-theorem/