# Aspect/Innsbruck Interpretation which respects SR locality

1. Jan 23, 2005

### Hans de Vries

--
--

EPR experiments seem to show a significantly higher
correlation rate in the detection of separated photons which
are in an entangled state. From the measured correlation we
may or may not want to draw very fundamental conclusions.

One such a far reaching conclusion would be that our world
is fundamentally non-local and that "action on a distance"
is possible. This would be in serious friction with the Special
Theory of Relativity.

Rather then saying that QM predicts non-locality we need to
be more specific and state that the correlations measured
predict non-locality, that is, if all other alternative local
explanations are exhausted.

The most successful Quantum Field Theory, The Standard Model
which unifies the Electromagnetic, Weak and Strong forces does
not need "action on a distance". Path integrals do not "jump
space" and respect Special Relativity.

Some of the champions of the Standard Model have a strong
preference for local theories. For instance Gerard ‘t Hooft:
http://www.phys.uu.nl/~thooft/quantloss/sld020.htm

--------------------------------------------------

I would like to show an example of how a purely local
interpretation can give a much higher correlation equal
to the results of the Aspect and Innsbruck experiments
without the need for action on a distance.

I will discuss first the “Bell Inequality” case, the non-local
QM case and then the local alternative. I’ll use an example
based on the Wollaston Prism which is used in most if not
all EPR experiments.

The Wollaston Prism splits a light beam in two beams, one
horizontally and one vertically polarized. A single photon
is said to exit at either the horizontal or vertical output, but
never at both.

--------------------------------------------------

We will look at the case where the two entangled photons at
A and B are both polarized at 45 degrees with respect to the
Wollaston prisms:

The Bell inequality case: It is presumed that the photon at
A has a 50:50 % chance to exit at the horizontal or vertical
output and the same is true for the photon at B. However,
the outcome at A and B are presumed to be completely
independent even though the particles are entangled.
The correlation is calculated to be 50%

The non-local QM case: It is presumed that the photon at
A has a 50:50 % chance to exit at the horizontal or vertical
output. However when it exits at for instance the vertical
output then “action on a distance” causes the photon at B to
be also vertically polarized as a result of the measurement at A.
The correlation is assumed to be ~100%

The alternative local model: We presume that both photons
share a property because they are entangled. They are more
equal then other seemingly equal photons. If the photon at
A leaves at the horizontal output then B will generally also leave
at the horizontal output because they share this property.
Although different photons will exit at different sides, entangled
photons will typically leave at the same side resulting in a
correlation of ~100%

--------------------------------------------------

This would mean that the selection process at the Wollaston
Prism is not entirely random anymore but became predeter-
mined by the property at the place where the photons became
entangled.

This then requires a property to be explained. One possibility
I came across stems from the fact that fundamental photons
(spin 1 bosons) are either left or right circular polarized.
So called linear polarized single photons as presumed in the
EPR experiments can not be fundamental since they would
have spin 0.

Linear polarized photons must be considered to be a combination
of a photon with spin up and a photon with spin down. This now
introduces extra degrees of freedom. These degrees of freedom
may be random for arbitrary photons but equal for entangled
photons coming from a PDC.

The particular constitution of the up and down photon may make
the difference in the birefringent beam splitter where a choice
is forced for the 45% polarized combination to exit at either the
horizontal or vertical polarized output.

--------------------------------------------------

This example just goes to show that one should exhaust
all possible local explanations before such far reaching
conclusions as non-locality can be made with certainty.

It’s my opinion that the above alternative should be
disproved convincingly in order to prove non-locality.

Regards, Hans

PS. The difference in correlation in the actual Aspect and
Insbruck experiments is not as high as here stated because
several things are going on at the same time. The photons
are assumed to be superposition states and various different
angles are used randomly (0, 22.5 45 and 67.5 degrees)

2. Jan 24, 2005

### vanesch

Staff Emeritus
The statement that EPR experiments indicate that "quantum mechanics is inherently non-local" is over-stretched. The unitary part of quantum mechanics is (as you point out, in the standard model for example) perfectly local. What is non-local is the projection postulate, when applied too early. In a many-worlds like context, nothing non-local is going on, and this explains also in a quite natural way why we can "see correlations which point to nonlocality after the fact" but that we can never observe them "in real time" and have to wait for the sub-lightspeed signal in order to find it out.

So if you take the attitude that QM only predicts statistics, then there is not really a problem, because nothing "inherently nonlocal" happened. The only thing that is disproved by Bell's inequalities is that these statistics cannot be generated by an underlying local realist model.
If you take the attitude that QM describes physically what happens (my preference) and take the Hilbert space as something physical, then what goes wrong is the collapse at a distance: so simply don't collapse, and you're ok too.

cheers,
Patrick.

3. Jan 24, 2005

### vanesch

Staff Emeritus
I would like to add the following:
If the EPR situation hadn't been motivated by Einstein's viewpoint, and successively by Bell's work, then the experiment could be interpreted in a totally different way, being a "proof" of macroscopic entanglement and not a "proof of non-locality".

Let us imagine that Einstein never insisted on an underlying mechanism that would generate the statistics (hidden variable model) - or for that matter, that Einstein wasn't such a heavy-weight in physics, and that Schroedinger was the "big guy" to whom everyone was paying attention.
Schroedinger (and his cat) claimed somehow that you couldn't have macroscopic superpositions of classical situations "because of the absurdity of the idea" (a cat which is dead and alive at the same time). In fact, Schroedinger understood very well that the unitary evolution of QM automatically lead to entanglement of macroscopic systems.

And let us imagine that an experiment would then be set up to show that even people can be in macroscopic superpositions. How would one try to do that ? How could we verify that Alice is in an entangled state ? The simplest way would be to entangle Alice with a microscopic system which is entangled with another system, of which we know it cannot influence Alice, because it is miles away, in a bunker 50 m underground. And in order to check whether Alice is really entangled, you let her come down later into the bunker, and see if she can interfere with that second microscopic system.
So what do you do ?
You generate 2 entangled particles (say, 2 photons), and "keep one in the bunker" and send off another one to Alice, miles away.

So we have:
|photonspins> = (|bunkerz+>|farawayz-> - |bunkerz->|farawayz+>)|alice0>

(this state is such that we can take z to be any direction)

She "measures" something about the faraway photon (say the spin component in the z-direction), which comes down to entangling her, locally, with that spin component. So we have now Alice in an entangled state:

|bunkerz+>|farawayz->|alicez-> - |bunkerz->|farawayz+>|alicez+>

This is the kind of state Schroedinger objected to. Note that to obtain the state, there has only been a LOCAL interaction at Alice's place, between her and the faraway photon.
|alicez-> is the Alice who saw a z- state, and |alicez+> is the Alice who saw a z+ state.

Now, how could we find out whether this superposition is total nonsense or not ? Let us let Alice INTERFERE with the state of the bunker photon !

So the two states of Alice now take the train, and come down to the bunker. In doing so, she gets extra correlated with a lot of environment, like the guy in her compartment to whom she shows her notebook etc... So you can now take on the situation in which |alicez-> stands in fact not only for alice, but also for the guy in the train and everything which gets correlated with her. It doesn't matter.

If Alice now comes into the bunker, and she decides to measure the spin of the other photon, which is still whirling around in optical fibers, if she's in an entangled state, she will get an interference result which shows in her correlations between her notebook of the faraway photon and the local bunker photon.
So Alicez- will work with the |bunkerz+> state and find accordingly the correct correlations, while Alicez+ will work with the |bunkerz-> state and will also find the correct correlations. So in both cases, Alice will realize she finds correlations with the state she's supposed to find *as if she collapsed the wavefunction* at a distance during her first "measurement". But in fact, her two states interfered with the bunker photon ! And of course, locally in the bunker, nothing "collapsed" when she did the other measurement !

So this experiment, which shows interference between the Alice states in her notes, shows us that we have to take the macroscopic superpositions seriously. This could have been the aim of the experiment.

If you take quantum mechanics seriously, you don't even need the Bell inequalities to prove the point. The very fact that Alice, if she measures the z-components everywhere, always finds perfect anticorrelation, indicates already her superposition. This is the outcome when you stay within the framework of QM, but you wonder whether it makes sense to talk about macroscopic superpositions.

But you then might find the objection that *maybe we don't know, but the photons DO carry hidden variables with them giving you the outcomes for the spins*. And in fact, the original "entanglement" of the two photons just meant a correlation of the hidden variables, like the classical example with a red and a black marble in two bags: if you know that there is a red marble in YOUR bag, you automatically know that there must be a black marble in the other bag. Note that saying that already gets us out of the scope of QM, because it means that the initial entanglement of the two photons is just an expression of a correlation of underlying hidden variables. It is at that point that Bell's inequalities show us that for certain combinations of choices of measurement by Alice, you cannot find hidden variables which could generate you the same correlations in her notebooks.

You can even increase the power of the experiment in proving that macroscopic entanglements must exist, by using 2 human beings, Alice and Bob, who measure simultaneously some spin components: Bob in the bunker, and Alice far away, and then they both take the train and meet in some intermediately located railway station.
Then it is the entangled Bob state which interferes with the entangled Alice state, and out come a superposition of "bob and alice" product states which have correct notes.

At no point in this discussion, locality has been put in doubt. What has been studied is whether macroscopic superpositions make sense, and the experiment gives strong hints that yes, you have to take macroscopic superpositions into account, if you do not want to "collapse wavefunctions" at 50 miles, 700 meter underground, in a bunker, and of which we never got any theoretical nor experimental indication.
Of course, if you INSIST on that collapse because you cannot stand the idea of macroscopic superpositions, you get into all kinds of weird ideas,such as non-locality, signalling back in time, and so on. But if you accept QM AS IT IS, there is no problem.

cheers,
Patrick.

4. Jan 24, 2005

### gptejms

Hans,
Given my little knowledge of QFT,I've the following question.
Could we assume the following picture for a linearly polarized photon:-it is a mixture of two one-photon states,one left circularly polarized and another right circularly polarized with equal probability amplitudes for both the states?
If yes,then the following question arises:-if such a photon is passed thru a polarizer how do you calculate the probability of it passing thru a polarizer?Do you first calculate the prob. amplitudes for the two states passing thru the polarizer,add them and then square to get the probability or you calculate the probabilities for the two states and simply add them?
The former looks to me the right thing to do.If yes,one needs to re-do the EPR calculations and see what we get.

5. Jan 24, 2005

### vanesch

Staff Emeritus
If you only want Hans to reply, then ignore my message

Ha, you hit the nail on its head. The question is, again and again: when do we use the Born rule ?
If you consider the first case, namely you calculate the coefficient of the vector component that gets through the polarizer, then you postpone the use of the Born rule - and that's what you should do.
If you apply the Born rule BEFORE the polarizer, then you calculate the individual probabilities.

Let's have a look: we take right-polarized photon |r> ; left : |l>
we take linear x-polarized light: |x> and y-polarized light |y>

we have that |r> = 1/sqrt(2) ( |x> + i |y>)
and : |l> = 1/sqrt(2) (|x> - i |y> )

You can inverse easily these two linear equations:

|x> = 1/sqrt(2) (|r> + |l> )
|y> = - i /sqrt(2) (|r> - |l> )

Let us assume we have an |x> state. You consider somehow the r/l base as more fundamental, so we say that |x> is a superposition of |r> and |l>.

probability to be in the |r> state: 1/2 and probability to be in the |l> state: 1/2. That's applying Born's rule, and you transform in fact the superposition into a statistical mixture.
But this goes terribly wrong !
Namely, what's the probability of an |r> state to be x-polarized ? It is 1/2.
Same for an |l> state. So how much of our original |x> state gets through an x-polarizer ??
1/2 x 1/2 + 1/2 x 1/2 = 1/4 + 1/4 = 1/2

So x-polarized light only gets half through an x-polarizer ??? ERROR !

No, you have to work with the Hilbert states. An |x> state is an eigenstate of the polarizer measurement, which has eigenvalue 1 for |x> and eigenvalue 0 for |y>. So an |x> state, being an eigenvector, will get through it 100%.

cheers,
Patrick.

6. Jan 24, 2005

### gptejms

Thanks vanesch for your imp. input.
Coming back to Hans:-it's amply clear that for a state like |x>,the prob. amplitude for the photon to pass thru a polarizer directed along |x'>, at an anlgle $$\theta$$ to |x>,is $$<x'|x> = \cos \theta$$---the probability is $$\cos^2 \theta$$--i.e. Malus' law is satisfied in this quantum scenario.For the classical case,treatment as in posts in the thread 'Bell's theorem and negative probabilities' stands.So nothing new really arises by considering left/right circular polarizations of a photon.

7. Jan 24, 2005

### DrChinese

1. I agree: this is actually exactly what the Bell Theorem leads us to... but many folks cannot see how we could get the correlations without locality being violated. So they assume non-locality is observed. But that is not strictly a consequence of Bell. Reality - as we commonly understand it - may not be an accurate way to look at the universe. I.e. it's a local non-realistic universe rather than a non-local realistic universe.

2. I do not follow you here. Bell clearly contemplated that local reality (LR) yielded identical predictions to QM, not different ones.

3. I couldn't follow your logic on this one. As I understand it, you are saying that we are measuring something different than a spin 1 particle. But how does that make the Bell Theorem conclusion - no local hidden variables - disappear?

8. Jan 24, 2005

### vanesch

Staff Emeritus

Right ! The issue is not locality, but "realism" (in the classical sense). But if you push in realism by all means, then it looks like locality is violated, causality is violated and all this in such a way that... you cannot really violate causality or locality

cheers,
patrick.

9. Jan 24, 2005

### DrChinese

Yes, I am glad you said that. I think it is also interesting that the same kind of spooky actions that happen in an EPR setup - i.e. instantaneous collapse of the wave function - also happen in plenty of other situations in which locality isn't an issue. Presumably, covering up one slit in a double slit experiment immediately changes something at the other slit. (And no one has a cow about that :rofl: even though it is no different.) So the point is that EPR tests really show nothing surprisingly new, they just show it so well!

10. Jan 24, 2005

### Caroline Thompson

In view of the various loopholes in the experiments, whether or not the observed correlation is incompatible with local realism is debatable. I'm interested, though, in your point about Wollaston prisms not necessarily splitting even light polarised at 45 deg 50-50. I'm beginning to accumulate evidence, not so much from EPR experiments as from other quantum optics ones, that the behaviour of the light may be influenced by the phase difference between the horizontal and vertical components. This applies only when the source is a PDC one, but in practice this means nearly every recent quantum optics experiment.

I'm afraid I have little use for the "photon" concept, or for the idea that plane polarised light is always formed by the addition of right and left circularly polarised waves. PDC outputs are sometimes polarised exactly vertically or exactly horizontally, but sometimes, I believe, both at once. In these cases, the phase difference between the two components is undeniably critical to the way the light splits at a Wollaston prism. The matter is mentioned in a footnote to Weihs et al's report of their 1998 Bell test at Innsbruck -- it comes from the standard classical theory of light. [I discuss how this may explain Weihs' results in http://arxiv.org/abs/quant-ph/9912082.] [Broken]

There may, though, be other subtle properties of the light and of the prism that influence the proportion in which the energy of an individual pulse of light it is split. The prism is actually a pair of prisms, with carefully-engineered layers of dielectric or metal, of thickness 1/2 or 1/4 wavelength, between the them. The behaviour of the light may well be influenced by its exact wavelength, with 50-50 splitting occuring only when that wavelength exactly matches the one for which the prism was designed.

No time for more ...

Caroline

http://freespace.virgin.net/ch.thompson1/

Last edited by a moderator: May 1, 2017
11. Jan 24, 2005

### DrChinese

This argument deftly diverts us from the essence of the experimental results (which do not support local realism and do support the predictions of QM). If there was no correlation observed, this might be meaningful. But since we see strong correlations, this issue is totally meaningless. It doesn't matter what kind of splitter is used as long as efficiency is high. And it also doesn't matter is there is some anisotropy. There cannot be "accidental" agreement due to a local polarizer bias in this kind of setup.

12. Jan 24, 2005

### Hans de Vries

In a similar sense I've got the feeling that, If Einstein, Podolsky and
Rosen would have been familiar with the Fourier Interpretation of
Heisenberg's Uncertainty Relation, we not would have the hidden
variables so much in the in the center of the discussion.

From the EPR paper on DrChinese's website:

So it's either (1) Hidden Variables or (2) No "simultaneous reality"

Regards, Hans

13. Jan 24, 2005

### Hans de Vries

The complication I see is the nature of the birefringent Calcite of the
Wollaston Prisms. It has a different diffraction index for horizontal and
vertical polarized light. (nE = 1.4864, nO = 1.6585) which means that
the speed of light depends on the polarization of the photons. Now
what will be the effect on a circular polarized photon (spin up or down) ?

http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/cdopt.html

Regards, Hans

14. Jan 24, 2005

### vanesch

Staff Emeritus
This is classical optics
Remember the quarter wavelength plates which transform linearly polarized light under 45 degrees into circularly polarized light ?

It is quite straightforward:

If |pol> is your incoming polarization, then you write |pol> in the x-y basis:
|pol> = a |x> + b |y>

|x> will follow the evolution with a refractive index n_x, and |y> will follow the evolution with a refractive index n_y.

This means that you can calculate how many wavelengths w_x have been gone through for the x-part (from the incoming surface of your optical device to the outgoing surface), and idem for w_y. Each component will then get a phase factor equal to exp(i 2 pi w_j), so the outgoing polarization is:

|pol-out> = a exp(i 2 pi w_x) |x> + b exp( i 2 pi w_y) |y>

( I use here QM notation, but the calculation is identical in classical optics of course).

Exercise: pure |x> will remain pure |x> and pure |y> will remain pure |y> (both get a phase factor: some optical path length).

Exercise: 1/4 wavelength plate:
incoming linear polarized under 45 degrees:
|pol> = 1/sqrt(2) (|x> + |y>)

quarter wavelength: w_x = n ; w_y = n + 1/4

thus: |pol-out> = 1/sqrt(2) ( |x> + i |y>)

circularly polarized light (due to the i)

Exercise: 1/2 wavelength plate:
incoming linear polarized under 45 degrees:
|pol> = 1/sqrt(2) (|x> + |y>)

w_x = n ; w_y = n + 1/2

|pol> = 1/sqrt(2) (|x> - |y>)

linearly polarized light which is 90 degrees rotated with incoming polarization!

Optics is fun.

cheers,
Patrick.

15. Jan 24, 2005

### Caroline Thompson

Quite. This classical optics is what I was talking about.

Caroline

16. Jan 24, 2005

### caribou

You just reminded me of something Murray Gell-Mann and Jim Hartle said about EPR. Here's the quote, the italics are theirs.

That's from their paper, Quantum Mechanics in the Light of Quantum Cosmology, the first paper to describe their Decoherent Histories work to clarify quantum theory. It's not freely available online but I have a scanned copy.

If you haven't read their work on this already, then I think you'll find it very interesting. Hartle introduces some parts of their work here in a different paper that talks about the two-slit experiment:

http://arxiv.org/abs/gr-qc/9210006

17. Jan 24, 2005

### JesseM

How is "locality" defined in the context of quantum theory? This isn't obvious to me if you represent a system's state as a vector in Hilbert space. For a classical field, each point in space is assigned a force vector, and "locality" basically means that the direction and magnitude of the vector at a given point in spacetime depends on the vectors at all the points in its past light cone, but does not depend on the vectors at points outside the past light cone. Can a system's quantum state be described by assigning some mathematical entity (analogous to the force vector) to each point in space, with "locality" having the same sort of meaning?

18. Jan 24, 2005

### vanesch

Staff Emeritus
The Hilbert space state is of course not local, in the same way as a point in classical phase space is not "local". By local is meant only that physical processes only work on the components of the tensor product of spaces which correspond to systems which are local.
So if we have a general state in the space H1xH2xH3 where system 1, described by H1, is far away, and H2 and H3 are systems in eachother's neighbourhood, then the general state:

a |u1>|u2>|u3> + b |v1>|v2>|v3>

undergoing a local physical process involving H2 and H3
can only change |u2>|u3> into something else and |v2>|v3> into something else, according to the same unitary transformation U23.
Locally at "23" you cannot do anything about u1 or even know anything about it ; it is completely transparant to the physics at "23".
I think this is what is generally meant with locality: an interaction can only influence the part of the description which corresponds to local systems.

cheers,
Patrick.

EDIT: I should point out maybe that what I think is understood by "locality" is "the ability to influence, through interaction" and not "the description of the state of the universe". In fact, in pre-relativistic physics, locality might have been a philosophically satisfying, but not strictly necessary requirement. It is relativity which makes "locality" equivalent to "causality". And causality is a much harder thing to sacrifice.
But all this has only to to with *interactions*. So as long as interactions only influence that part of the state description which has to do with the locally available systems, I think one can conclude that the theory is "local" in the sense required by relativity and causality.

So in quantum theory, the state description is not local, but the interactions satisfy locality, at least as long as one doesn't apply the projection postulate. The projection postulate is bluntly non-local ; so if the projection postulate is part of the possible sets of interactions in nature, then there is a real problem with causality ; and it is THIS thing which people are always talking and writing about. But - as I tried to point out several times - you don't need to apply that postulate non-locally: you can wait until you arrive at your final, local, measurement of the correlations (after Bob and Alice came together to compare their notebooks). And if you do so, all the mystique of the non-locality of QM drops. The mystery is not so much in the fact that it could have been non-local (and hence non-causal), but in that the inherent mechanism in QM is such that it is non-local and non-causal, but that each time you would like to USE that non-locality to make a faster-than-light phone, somehow there is a conspiracy in the theory that forbids you to do so.
If you understand that there is also a non-a-causal explanation to the EPR experiments (with Alice and Bob in entangled states), then this "conspiracy" is easily explained.

EDIT2:
the fact that you want to assign numbers to each local system (for instance, through fields) which will then determine completely (deterministically or stochastically) what will happen to that local system, I would call that "realism", in that there is a complete description of the local system which corresponds to its "real behaviour".

For instance, Newton's theory of gravity is realist but non-local (the "real state" of each particle is locally described by its velocity and (of course) its location) but gravity works at a distance.
Maxwell's electromagnetism is realist and local (all you need to know locally about the state of the EM field is given by the local values of E and B ; and they influence eachother only through their derivatives).
Quantum theory is non-realist but local.

Last edited: Jan 25, 2005
19. Jan 24, 2005

### vanesch

Staff Emeritus
A more formal answer: yes, in QFT, field operators which correspond to spacetime events which are space-like connected, commute. This is the requirement of locality in QFT. It comes down to what I said in my previous post: an interaction, depending on the field operators at A, will not influence any component of the state vector at B.

cheers,
Patrick.

20. Jan 25, 2005

### Hans de Vries

Let's see. are you assuming here that for the case of 45 degrees Bell
and entangled QM should be equal? That is, in the electron LR theory
of Bell?

The two differences here are
1) is that were looking at photons.
2) is that both a and b have equal angles.

So we have the 0 degrees case here. See figure 4 of Aspect
http://www.drchinese.com/David/EPR_Bell_Aspect.htm
and figure 3. of Caroline's paper which shows the idealized QM and LR
curves. http://arxiv.org/abs/quant-ph/9903066

If a linear polarized photon is a composition of a spin up and a spin
down particle then there are extra degrees of freedom to combine
them. These extra degrees of freedom may be random in general but
equal in the two entangled particles.

The extra degrees of freedom may also influence the outcome of the
measurement and thus increase the correlation well beyond what is
shown in figure 3 of Caroline's paper for the LR case.

Caroline's calculation for the LR case assumes that the only thing they
have in common is the polarization angle.

Regards, Hans

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