# Consequences of Hardy's paradox and joint weak measurement

1. Mar 11, 2009

### analogdigital

I read two articles recently and I am wondering if I have the correct interpretation of them.

They deal with joint weak measurement and Hardy's paradox.

Is the result of the "Direct observation of Hardy's paradox by joint weak measurement with an entangled photon pair" paper indicating that the wave function is, indeed, already 'collapsed' before we look? That is, does an objective reality already exist, at least in terms of probability? Or does this only point out that statements about past actions can now be made without disruption of the system? I am slightly confused about a few aspects of this. Another is, does this prediction of these past events mean that predictions about the future of other observations is also then possible, reducing the number of future realities to one, or at least one most probable? Or does the indeterminacy still apply and we only have a probable result, which can be affected by measurements and observation?

Summary article: http://www.sciencedaily.com/releases/2009/03/090304091231.htm

Full paper: http://www.iop.org/EJ/article/1367-2630/11/3/033011/njp9_3_033011.html

Second article:
http://www.scientificblogging.com/n...hey_have_resolved_hardys_annihilation_problem

A response about both would be great, as they have different methods of attaining their results.

I still don't see how this overcomes HUP. I understand that the amount of uncertainty required is larger than the presence of the instruments for detection, but I don't see how this really resolves anything in theory. I can see the practical applications, but I'm foggy as to what this clears up or lends itself more towards. (decoherence, many worlds, Copenhagen, etc) Any help would be appreciated greatly!

Thanks! First-timer poster here :)

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2. Mar 11, 2009

### DrChinese

Welcome to PhysicsForums, analogdigital!

You started off with a pretty complicated subject, and understanding the results of these experiments is tricky. I don't want to pass myself off as an expert on these type experiments, but I may be able to get the discussion going at least.

You get Hardy's Paradox (0=1) when you assume a realistic formulation over and above standard QM. So I would not consider the resolution of the paradox - in which you will get a specific value - to support the realistic view.

The experiment can be considered a way to determine "which-path" information to a variable degree. You get 0 at one extreme (maximum which-way), 1 at the other extreme (no which-way), and a mixture in between according to Figure 2. So to me, the HUP is followed because you would expect such a tradeoff. The paradox is resolved because you see that the results are dependent on the strength of the measurement.

Keep in mind another characteristic of entangled photons: they lose their interference characteristics if which-path information can be determined. (This was known before this experiment, and had been demonstrated in a variety of contexts. If this were not true, it would be possible to construct FTL signalling devices.) You may have known this before, but I personally think this experiment shows this characteristic nicely.

3. Mar 11, 2009

### analogdigital

When you say realistic formulation, are you referring to a reconciliation of "spooky action"? (Basically, are you saying when I assume an objective existence/probability outside of one that's measured?) And could you clarify what you mean by "standard QM"? From what I can tell, these experiments tether the ability to predict future results to a level of reasonable probability. This makes me question indeterminate aspects of QM. Perhaps I'm missing something vital here.

When you say the HUP is followed, doesn't this imply that the weak measurement actually DID disturb the system and these results only produce something as contradictory as something like the dissonance that STCW and SEW seemed to imply when we accepted that Pwv() is not definite? Doesn't this go against these experiments being of quantum nondemolition type?

Does this mean that the particles aren't really in two places at once after ANY measurement is made, even weak? If not, what of the negative weak values? Couldn't we say that this method can only make predictions about what we measured, not would happen if we measured something else?

Any chance for a quantum eraser type of interaction where the interference could be restored after the weak measurement?

Sorry for all the sporadic weaving of questions, just trying to reason away what seems to be unreasonable! Any personal ideas on which interpretation of QM seems to be the most legitimate according to experimental data thus far?

4. Mar 11, 2009

### DrChinese

I will do my best, not sure if I can cover all of the ground satisfactorily - especially in one post.

1. Hardy's Paradox results from making a reasonable (read: realistic) deduction about the particles' activities when they are allowed to take a variety of path options. This "reasonable" deduction essentially forces you to admit X=0 and X=1 simultaneously. But you wouldn't make that deduction if you knew X=0 or X=1, or X=.5 for that matter, because there was another variable Y in the equation. Y being, in this case, the strength of the measurement and therefore the amount of "deduction" required. The less deduction, means stronger measurement (Y), also means more which-way known, also means less interference effects, which in this case is constructive. ("...photons 1 and 2 simultaneously arrive at BS3, due to a two-photon interference effect, they always emerge at the same port...") So that means lower visibility (Fig. 2) when the measurement is stronger. (Unless, of course, I reversed a few things along the way )

So to me, the above translates: the assumption of realism is only valid (reasonable) when you test for that attribute specifically. Otherwise, assumption of realism is invalid.

2. I would say that the weak measurement does disturb the system. The weaker the measurement, the less the system is disturbed. Either way, I believe the HUP is followed (respected) in the results. The stronger the measurement, the more there is demolition, and this shows in the results too. As you learn about the which-path, you make the photons act more like a particle and less like a wave. And that has the effect of reducing the interference effects.

3. I think - not 100% sure in case that wasn't clear - that the stronger measurements mean that you are gaining a kind of which-path information. And therefore the photons are not going through all possible paths. And therefore the interference effects change.

To put that on a par with DCQE experiments: try this link: http://grad.physics.sunysb.edu/~amarch/ [Broken]. They discuss how which-way affects interference, both pro and con.

4. To answer your question about viable quantum interpretations: they are all valid in various ways, each offering trade-offs. It is worthy to note: Yakir Aharonov, one of the authors of key reference [3] in the paper cited, has written about the Time Symmetric interpretation. If you haven't read this already, I would recommend it.

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5. Jun 11, 2009

### analogdigital

Ok, so I've done some more homework on this. Could anyone tell me their view on the potential ontological implications of these results? They seem philosophically significant to me. This leads me to believe that the paradox exists, which runs counter intuitive to fixed-ontological thinking. i.e. "The path is determined independent of observation. Although it has not happened yet, we can predict it with accuracy, as well as look at the results subsequently, without disruption, and determine our accuracy. However, there are two antithetical entities in occupation of the same space and time that should destroy each other, but don't. Therefore, the paradox exists and empirical contributions to the field of ontology are only probabilistic, not certain."

6. Aug 11, 2009

### Demystifier

Analogdigital, for my opinion on the ontological meaning of weak measurements see
https://www.physicsforums.com/blog.php?b=1225 [Broken]

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7. Aug 12, 2009

### Demystifier

https://www.physicsforums.com/blog.php?b=1226 [Broken]

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8. Aug 14, 2009

### Descartz2000

(original comment posted on weak measurement post) - How does Bohm's model explain the two particles that meet in the annihilation area and do not destroy each other? Are you saying the antiparticle measurements do not really reflect anything in the real world and is a construct of mathematics, and the existing particle is only in one arm the whole time?

Last edited by a moderator: May 4, 2017
9. Aug 24, 2009

### Demystifier

10. Aug 24, 2009

### Descartz2000

So, it sounds like the empty part of the wave function is real, but that it has no influence on any aspects of the experiment, nor does it leave any observable trace of itself. Therefore, there can be no interaction and ultimately no annhilation? Is this correct according to Bohm?

11. Aug 24, 2009

### Descartz2000

Demystifier,
I am still not sure why the Bohm interpretation states that an annhilation can not occur between the electron and the positron?

12. Aug 31, 2009

### Demystifier

But I have explained everything in the post #14 above. Can you specify what part of it you don't understand?

Or let me rephrase. The Bohm interpretation does not say that annihilation does not occur. Instead, it says the following: IF the calculation of the wave function in the mentioned paper is correct, THEN the annihilation does not occur because electron and positron do not interact when they meet each other.
However, irrespective on the interpretation, my point is that the calculation of the wave function in the mentioned paper is NOT correct.

13. Feb 9, 2010

### analogdigital

So what should it be? Also, do you know of any more recent experiments that contradict this, or is this to be considered conclusive?

14. Feb 10, 2010

### Demystifier

I have not yet made a correct calculation, but I plan to do it in the future. An experiment with electron/positron pairs have not yet been performed.

15. Feb 12, 2010

### Demystifier

OK, now I have done some more careful thinking of that stuff, done some calculations, and compared it with other calculations and discussions in the literature. Here are my conclusions.

In the ORIGINAL Hardy paper all the calculations are correct. What is even more important, Hardy clearly and correctly concludes that there is a paradox ONLY if one assumes that reality is LOCAL. He clearly and correctly states that nonlocal reality, such as Bohmian interpretation, resolves the paradox. Therefore, I think it is safe to say that, if nonlocality is allowed, then there is no paradox at all. Some later authors (such as Aharonov et al) find a paradox because they tacitly assume that the description should be local.

In fact, the point of the original Hardy paper is not that there is a paradox (because there isn't), but that such nonlocal reality is not compatible with Lorentz-invariance. However, this conclusion of the Hardy paper is wrong. He correctly concludes that there must be some particular Lorentz frame which defines the actual behavior of nonlocal reality. However, from this fact he incorrectly concludes that such a particular Lorentz frame must be "preferred", i.e., that it violates Lorentz invariance. As I stressed many times, the particular Lorentz frame may be determined dynamically, through relativistic covariant equations of motion and particular initial conditions. Of course, even in local classical mechanics, initial conditions are never Lorentz invariant, but it does not change the fact that the theory is Lorentz covariant in the sense that the equations of motions are Lorentz covariant.

16. Feb 12, 2010

### Demystifier

What Bohm interpretation says is that electron and positron (viewed as pointlike particles) do not interact directly through a contact interaction, but indirectly through a common wave function that guides them. Thus, under certain circumstances it is possible that positron and electron come into the same position at the same time, without being annihilated. For this to happen, the wave function must take a particular form, such as the Hardy state.

17. Feb 15, 2010

### Demystifier

The Hardy setup is presented as a paradox in the Aharonov et al (2002) paper. The implicit assumption in that paper is that an electron (or a positron) is one object. That assumption is the origin of the paradox. With this insight in mind, it is trivial to understand how the Bohmian interpretation avoids the paradox. In the Bohmian interpretation, an "electron" is not one object, but consists of TWO objects: wave function AND particle. Thus, it is easy to understand how can electron both "be" and "not to be" at the same time at the same position: The wave function is there, but the particle may not be.

18. May 11, 2010

### Demystifier

Let me present some additional insights on the "Hardy paradox" (which would be better to call Aharonov et al paradox, because it was Aharonov et al who first argued that there is a paradox in the Hardy set-up).

After reading the Aharonov et al argument again, I have identified the precise location of their mistake. Their mistake can be reduced to a purely logical mistake. To understand more clearly what sort of the logical mistake their mistake is, let me present a simple example of the logical mistake of the same sort. Consider the statement:
"Assume that it is raining. If I don't have an umbrella, I will get wet."
At first sight, such a reasoning seems perfectly logical. Nevertheless, it is incorrect. It is possible that I don't have an umbrella and yet that I will not get wet, provided that I have some other sort of a shield above my head. The umbrella is not the only possible object that may prevent getting wet. So more correct statement would be
"Assume that it is raining. If I don't have SOMETHING (that can be put above my had), I will get wet."

Now to the point. I will not repeat the whole argument of Aharonov et al, but only present the crucial step in their argument which is wrong. At some point they argue (I'm not using their exact words, but the meaning is the same):
"If detector D+ clicks, then electron took its overlapping path".
But this is wrong. The correct statement is
"If detector D+ clicks, then SOMETHING took its overlapping path".
But what that "something" may be if not the electron? In classical physics - nothing. But in quantum physics, it may be (and actually it is in the Hardy set-up) a COMMON wave function which describes the ENTANGLEMENT between the electron and the positron. More precisely, this part of the wave function which can be said to be in the overlapping path has the form
$$|+o\rangle |-no\rangle + |+no\rangle |-o\rangle$$
where "+o" denotes positron in the overlapping path, "-no" electron in the non-overlapping path, etc. It is this wave function (superposed with $$|+no\rangle |-no\rangle$$) that causes that D+ can click. And for this wave function it is not correct to say that the electron is in the overlapping path.

We also see that in the wave function above it cannot be the case (i.e., the probability is zero) that BOTH the electron and the positron take the overlapping path. (It also reflects in the Bohmian interpretation that it cannot be the case that the trajectories of electron and positron both take the overlapping path.) This clearly avoids the "Hardy paradox", because now we see how it can be the case that something IS in the overlapping paths such that both D+ and D- may click, and yet that it cannot be the case that both electron and positron are in the overlapping paths (which otherwise would annihilate if they were both there). That "something" which is there is a JOINT wave function, not separate electron and positron wave functions. When there is entanglement, there is no such thing as separate wavefunctions of individual particles.