Loopholes in optical Bell tests

[DrChinese]

Actually, if you properly calculate the correlation of the signals reaching the two detectors, a classical treatment also yields Malus' law for the angular dependence of the coincidence rate (see my page [link deleted] ), so it has hard to see what the whole point of the Bell test experiments is in the first place.

Thomas

Adding to ZapperZ's comments:

I read your page. If you believe in Malus' Law, then you are saying that you also agree that - per Bell's Theorem - there is NO LOCAL REALITY. It doesn't matter whether the QM predicted value is reached by your derivation or any other. If the QM value (which is of course also Malus' cos^2 theta) is accurate, LR is ruled out.

So the point of the Bell test is to rule out LR predictions which deviate from Malus. Don't be fooled that your classical derivation is somehow local realistic. It isn't! Bell forbids it because there are additional requirements you don't address.

So I hope your comment is not intended to imply that you have shown that classical treatment can lead to a local realistic outcome.

 

[vanesch]

If you believe in Malus' Law, then you are saying that you also agree that - per Bell's Theorem - there is NO LOCAL REALITY.

There always seems to be a confusion when people say that "QM predictions are in agreement with Malus' law". The trick being of course, that Malus' law needs a re-interpretation in the context of a corpuscular theory (photons).

Classically, Malus' law simply says how much intensity gets through a polarizer when we know the angle between the polarizer and the plane of polarization of the light: I(through) = I(in) x cos^2 (theta) where theta is this famous angle of course.
But as such, this doesn't say anything about correlations between individual particle detections, as is the case in these Bell experiments. We need an extra hypothesis that links the particle detection with "classical intensity", and of course the standard way of doing this is by assigning a probability of detection somehow proportional to the intensity of the beam (at low intensities).
But then I don't understand the claim that the QM predictions would be in agreement with Malus' law. Take the two polarizers to be perfectly parallel (say, parallel to the Y axis). We know from quantum mechanics that a singlet state will HAVE 100% CORRELATION (meaning, whenever the Bob detector clicks, Alice's detector clicks too and whenever Bob's detector doesn't click, Alice's doesn't, either).

But that is NOT what Malus' law predicts ! Taking that the incoming polarization of the photon can be just any orientation, the probability of clicking at Bob's is proportional to cos^2theta, and the probability of clicking at Alice's is also proportional to cos^2theta. BUT theta IS UNIFORMLY DISTRIBUTED !
Indeed, for theta 0 (the classical light is also parallel to Y), we have 100% efficiency to click at Bob, and 100% efficienty to click at Alice, so the correlation is indeed 100%. But for theta = 45 degrees, we have 50% chance to click at Bob, and INDEPENDENTLY 50% chance to click at Alice, which means that we only have 50% correlation in this case.

The overall correlation is given by the number of clicks AT BOB AND AT ALICE, over the number of clicks AT BOB.

clicks at Bob : \frac{1}{2 Pi}\int cos^2(\theta) d\theta = 1/2

clicks at Bob and Alice: \frac{1}{2 Pi}\int cos^4(\theta) d\theta = 3/8

which comes down to only 75%.

This is not the 100% (with identically aligned polarizers) QM predicts. So what does it mean that "QM has the same predictions as Malus' law" ?

Or is this a "perverse" application of Malus' law, where it is applied ONCE THROUGH THE PROJECTION POSTULATE we have jumped the polarization of Alice's photon into the polarisation given by Bob's polarizer when it clicks ?

 

[Sherlock]

My comments also relate to Rade's last post.

We agree that Bell's Theorem says nothing about the role of humans, and that is the interpretation of most. Bell's Theorem also does not mention the actual existence of fundamental particles depending on observation, something I am sure we agree upon.

Wrt Rade's last post, I'm not sure that he understands all that's involved here. I'm quite sure that I don't. :-)
In any case, I agree with all your statements, cited above and below, but I feel like elaborating. :-)

Talking about what exists independent of observation is necessarily less clear when dealing with quantum phenomena than when dealing with macroscopic phenomena. The EPR idea that we can take (at least wrt certain instrumental configurations) a measured value-property as being in one-to-one correspondence with something existing in nature independent of observation has been shown to be false when applied (via Bell and Bell tests) to most instrumental settings. But even before Bell and Bell tests it was evident that there's no well-defined (unambiguous and consistent) qualitative characterization of what a photon, *is* in nature. All we have is our macroscopic apprehension of instrumental results. So, the way that we talk about the photon (ie., the way that the results are modeled) depends in large part on how the results correspond to phenomena of our normal sensory experience (eg., waves and particles, which we can abstract and manipulate unambiguously and consistently). Wrt to some setups a wave charactarization is adequate, and wrt others a particle characterization is necessary.

Wrt Rade's concern, none of this means that reality, even at the level of quantum phenomena, doesn't exist when it's not being observed. It's just not too clear what reality at the level of quantum phenomena *is*. The term, photon, as defined in quantum theory, doesn't refer to something that exists in nature independent of observation. On the other hand, the moon for example does refer to something that exists in nature independent of observation.

Einstein wanted a theory of quantum phenomena that talked *directly* about those phenomena as they exist in nature. He regarded quantum theory as incomplete because it doesn't do that -- and quantum theory *is* incomplete in that sense. However, in learning about the development of the theory, I've come to the conclusion that it has been constructed in as complete and realistic a way as any theory could have been considering what it's dealing with.

Bell's Theorem only addresses whether particles have simultaneous well-defined local values for all quantum observables. They don't, and Bell tests prove this.

Ok, and it's good to remember that it's we humans who are doing the defining. We might conclude therefore that our idealized conception of polarization for example is not telling us all the relevant aspects of the behavior of the light incident on, and transmitted by, the polarizers -- or of the details of the relationship, if any, due to emission from a common source, between the light, associated with joint measurments, incident on A and B.

Another way of framing it is that a certain general form embodying the separability (assuming this to be the only allowable form consistent with a limiting c) of events at A and B is seen to not apply to most joint (A,B) measurements in Bell setups. So, either c isn't a limit, or the joint measurement context is a nonseparable one for some other reason. Maybe, in a universe where all transmissions are limited by c, situations in which two particles are emitted at the same time from the same atom, and are being analyzed jointly by the same sort of device, are nonseparable -- in which case, the defining characteristic of the form of a proposed lhv description would be nonseparability.

It does, I think anyone would have to admit, make sense to think of the combined motions of two separated particles which have interacted or have a common source as being related (ie., nonseparable) in some way. This is, after all, the basis for conservation equations. It's the basis for the qm treatment of such biparticle situations in terms of a single wave function.

But, the problem is the same as when dealing with any quantum process. There's no qualitative apprehension of what's happening at that level. As has been demonstrated, just drawing a line in a unit circle and saying that that represents a common polarization value doesn't quite cut it.

So, as I see it, whether there are nonlocal transmissions in our universe, and whether lhv theories are in principle possible are still open questions.

By all standards of what consists of knowledge, there are no loopholes. As ZapperZ pointed out previously, we are more sure about the results of Bell tests than we are about many things. If you dismiss Bell tests because of purported loopholes, you must dismiss much of what we know about physics.

There are loopholes and they're important for metrological reasons. But I agree with you that the Bell tests are pretty conclusive evidence that the inequalities are being violated and that the qm predictions are correct.

 

[Sherlock]

There always seems to be a confusion when people say that "QM predictions are in agreement with Malus' law". The trick being of course, that Malus' law needs a re-interpretation in the context of a corpuscular theory (photons).

Classically, Malus' law simply says how much intensity gets through a polarizer when we know the angle between the polarizer and the plane of polarization of the light: I(through) = I(in) x cos^2 (theta) where theta is this famous angle of course.
But as such, this doesn't say anything about correlations between individual particle detections, as is the case in these Bell experiments. We need an extra hypothesis that links the particle detection with "classical intensity", and of course the standard way of doing this is by assigning a probability of detection somehow proportional to the intensity of the beam (at low intensities).
But then I don't understand the claim that the QM predictions would be in agreement with Malus' law. Take the two polarizers to be perfectly parallel (say, parallel to the Y axis). We know from quantum mechanics that a singlet state will HAVE 100% CORRELATION (meaning, whenever the Bob detector clicks, Alice's detector clicks too and whenever Bob's detector doesn't click, Alice's doesn't, either).

But that is NOT what Malus' law predicts ! Taking that the incoming polarization of the photon can be just any orientation, the probability of clicking at Bob's is proportional to cos^2theta, and the probability of clicking at Alice's is also proportional to cos^2theta. BUT theta IS UNIFORMLY DISTRIBUTED !
Indeed, for theta 0 (the classical light is also parallel to Y), we have 100% efficiency to click at Bob, and 100% efficienty to click at Alice, so the correlation is indeed 100%. But for theta = 45 degrees, we have 50% chance to click at Bob, and INDEPENDENTLY 50% chance to click at Alice, which means that we only have 50% correlation in this case.

The overall correlation is given by the number of clicks AT BOB AND AT ALICE, over the number of clicks AT BOB.

clicks at Bob : \frac{1}{2 Pi}\int cos^2(\theta) d\theta = 1/2

clicks at Bob and Alice: \frac{1}{2 Pi}\int cos^4(\theta) d\theta = 3/8

which comes down to only 75%.

This is not the 100% (with identically aligned polarizers) QM predicts. So what does it mean that "QM has the same predictions as Malus' law" ?

Or is this a "perverse" application of Malus' law, where it is applied ONCE THROUGH THE PROJECTION POSTULATE we have jumped the polarization of Alice's photon into the polarisation given by Bob's polarizer when it clicks ?

Would a better way to show the difference between a characteristic lhv formulation and the qm formulation be in terms of expectation values between +1 (perfect correlation between results at A and B, Theta = 0) and -1 (perfect anti-correlation, Theta = 90 degrees), where the qm values plot a curve and the lhv values plot a straight line?

As for the applicability of Malus' Law -- just for fun:
Take the joint (A,B) measurement as a single, non-analyzable dependent variable.
Take Theta, associated with (A,B), as a single, non-analyzable independent variable.

Assume that A and B are measuring the same light.

Take 'intensity' to refer to the (A,B) coincidence count.

recorded 'intensity' = maximum 'intensity' x cos^2 Theta.

Ok, that was pretty perverse. :-)

But, I do think that part of the interpretational problem (wrt Bell's Theorem and a general form for lhv theories) lies in assuming that lhv theories must be constructed in a separable form.

 

[DrChinese]

There always seems to be a confusion when people say that "QM predictions are in agreement with Malus' law". The trick being of course, that Malus' law needs a re-interpretation in the context of a corpuscular theory (photons).

Classically, Malus' law simply says how much intensity gets through a polarizer when we know the angle between the polarizer and the plane of polarization of the light: I(through) = I(in) x cos^2 (theta) where theta is this famous angle of course.

...

which comes down to only 75%.

This is not the 100% (with identically aligned polarizers) QM predicts. So what does it mean that "QM has the same predictions as Malus' law" ?

Or is this a "perverse" application of Malus' law, where it is applied ONCE THROUGH THE PROJECTION POSTULATE we have jumped the polarization of Alice's photon into the polarisation given by Bob's polarizer when it clicks ?

The point is that for those who want to roll back the clock BEFORE QM arrived on the scene, you have to have Malus as your backstop. Otherwise, you are really in trouble because you are ignoring the classical experiments. (The same thing happens with those who deny special or general relativity, they must go back to Newton.)

So if you apply Malus with a local realistic bent, as you calc'd, you get perverse values for correlations. They are perverse, of course, because they are much further away from any value that you would ever measure in nature.

On the other hand, if you apply Malus with a quantum mechanical bent (which I don't consider perverse at all), i.e. as applying on a particle by particle basis, you get exactly the values that are measured in experiment. I.e. once you know Alice's polarization, Bob's matches the application of Malus. Why wouldn't it? After all, such application also matches the results for an ensemble of particles as well.

For the local realist, all this is a big problem. They now have Bell's Inequality to consider, and they have the QM predictions (which must be argued as being wrong even though they match experiment), and yet they have Malus (applied local realistically) which is really far off. So now the local realist must concoct yet another prediction which is much closer to Bell's Inequality if it is to be considered seriously (because the experimental error/loophole idea otherwise makes no sense because the deviation is way too large). Yet not a single local realist will advance such a prediction because it is absurd on the face of it.

Certainly the local realists know of the function you describe (which varies between .25 and .75), but they never stand by it. So once again I ask the local realist: what is the function that describes entangled photon correlations?

 

[vanesch]

On the other hand, if you apply Malus with a quantum mechanical bent (which I don't consider perverse at all), i.e. as applying on a particle by particle basis, you get exactly the values that are measured in experiment. I.e. once you know Alice's polarization, Bob's matches the application of Malus. Why wouldn't it? After all, such application also matches the results for an ensemble of particles as well.

Ah, you mean: the particles BEFORE measurement have/don't have a polarization, whatever, but ONCE Alice made her measurement, and this, by the projection postulate, MADE THE SYSTEM JUMP INTO ONE OF BOTH polarizations defined by Alice's polarization angle (parallel, or perpendicular, no other way out) and made, also through the projection postulate, JUMP BOB'S PHOTON ALSO IN THAT STATE (due to the entanglement), THEN, we apply Malus' law at Bob's side to calculate what is the probability is for him to get a click up or down according to the angle between the "polarization" (defined by Alice's measurement) and his analyser.
Yes, that's right, that gives the same result as the QM predictions of course, but I find this far-fetched to simply call this "Malus' law" because - that's what you call the quantum mechanical bent - there's the (non-local) projection postulate acting here before you apply it. My impression when people said "it's the same as Malus' law" was that it meant that using classical optics, you got the same results, which - as you point out, is totally wrong.

 

[DrChinese]

Ah, you mean: the particles BEFORE measurement have/don't have a polarization, whatever, but ONCE Alice made her measurement, and this, by the projection postulate, MADE THE SYSTEM JUMP INTO ONE OF BOTH polarizations defined by Alice's polarization angle (parallel, or perpendicular, no other way out) and made, also through the projection postulate, JUMP BOB'S PHOTON ALSO IN THAT STATE (due to the entanglement), THEN, we apply Malus' law at Bob's side to calculate what is the probability is for him to get a click up or down according to the angle between the "polarization" (defined by Alice's measurement) and his analyser.
Yes, that's right, that gives the same result as the QM predictions of course, but I find this far-fetched to simply call this "Malus' law" because - that's what you call the quantum mechanical bent - there's the (non-local) projection postulate acting here before you apply it. My impression when people said "it's the same as Malus' law" was that it meant that using classical optics, you got the same results, which - as you point out, is totally wrong.

My apologies for not making this clear. Your desciption is accurate. I was trying to show that at some level, Malus is common ground for the QM and LR positions. I absolutely believe that the projection postulate is to be applied as you describe. If it wasn't, then you could learn more about Bob that the HUP allows. That is really what EPR assumed would happen - that you could use Alice to learn about Bob. We now know that doesn't happen.

The problem from the local realist side is that for the case of unentangled photons, we still (also) have the cos^2 theta function to deal with. How does the local realist get cos^2 theta for a single unentangled photon and end up ignoring the cos^2 theta function (i.e. Malus) in the situation of entangled photons?

Of course, the local realist can say that there is no such thing as entangled photons. In that case, the probability functions are factorizable and presumably you MUST apply cos^2 theta separately. If you do that, you get the perverse 1/4 + (cos^2 theta)/2 function and this wildly deviates from experiment.

In other words, the local realist likes to think that the difference between experiment and Bell's Inequality can be accounted for by experimental error - i.e. the "loopholes". After all, there isn't a gigantic difference between the 2 even though they have easily been distinguished experimentally. But that really solves nothing for the local realist, since their alternative is even further away from experiment!

In my opinion: IF there is a consistent theory that makes a specific prediction; and that prediction is born out by experiment; THEN asserting there exist "loopholes" which cannot be measured is a specious argument. Once a scientist measures them, I will believe it. In the meantime, you MUST accept what the results of experiment tell us. Otherwise, there is no objective basis for believing anything in science.