David Bohm was one of the more interesting researchers in the field of the foundations of quantum theory. While a graduate student at Berkeley under Oppenheimer (after a short stay a Caltech where he was unhappy), he wrote, with David Pines, some early very influential papers on plasma physics. His textbook on quantum theory is a model for almost all of our modern quantum textbooks. And then, of course, there is the Aharonov-Bohm effect which shows how the vector potential can affect results even when the particles involved transverse regions where the ~~vector potential~~ magnetic field vanishes. It was supposedly while writing this textbook on quantum theory that he began to question the foundations of quantum theory and which led him to develop a non-local hidden variable theory for nonrelativistic quantum mechanics-a task whose supposed impossibility in turn led John Bell to formulate his famous inequality. Bohm, however, had consorted with the Oppenheimer crowd at Berkeley and got pulled into the whole “are you a communist” political mess and thus could not obtain a job in the United States, instead obtaining a job first in Brazil (on the recommendation of no one less than Einstein) and then in the United Kingdom where he continued his work in foundations and consorted with an assortment of interesting characters, including the Dali Lama and Jiddu Krishnamurti.

As you can see, I’ve always been fascinated by Bohm’s life. But what do I think of his hidden variable theory? It must be said right off the bat that Bohm constructed his theory more as a proof of principle than as the final solution to the foundational problems as he saw them (In fact it is probably best to say that Bohm did not believe there was a “final” solution as far as I can tell.) Well at various times in my life I have felt that Bohm’s theory was an intriguing direction towards understanding quantum theory. But as I’ve learned more and more from quantum compting, I’ve begun to think less and less of this theory. Indeed, I would say that quantum computing has taught me that there is something radical missing from approaches along the lines of Bohm’s non-local hidden variable theory. Quantum computing ruined Bohmian mechanics for me.

“Bah!” you say. What can quantum computing, which is obsensibly founded along the mantra “do not question quantum theory…accept it and rejoice at the splended information processing you can now do!” have to do with questions from the foundations of quantum theory, who are questions more associated with philosophy than constructive technological advance (yes this was a low blow to philosophy…sorry couldn’t resist)?

Well I would say it has pretty much everything to do with interpretations of quantum theory! Take you favorite interpretation of quantum theory. Now ask the question, how does this interprettion help explain to me why quantm computing is more powerful than classical computing. Whenever I do this for any of the interpretations, I find that I walk away even more appreciative of the weaknesses of each of the interpretations of quantum theory. For example, back to Bohmian mechanics. Now how does the idea that there is a pilot wave, or such, guiding the trajectory of a particle give us insight into why quantum computers can efficiently factor integers? Sure it seems reasonable that non-local hidden variable theories can be more powerful than local hidden variable theories, but why does the particular implementation of a non-local theory, as advocated by the Bohmian interpretation crowd, give us any extra insight into the power of quantum computing? Indeed, this is the crux of my problem: the more I learn quantum computing, the more I see it conected to the theory of computation. And the more I see it connected to the theory of computation, the less satisfying I find explanations such as “well it’s just a non-local theory”. Explanations such as that are like saying BQP is in PSPACE, so the power of quantum computing is obviously that of PSPACE. This leads to further weaknesses, I think, like the extreme wastefulness of non-local hidden variable theories in terms of their representation of the flow of classical inormation. I mean one of the astounding result of quantum computing is not that you can factor integers, but that you can’t also do everything in, say, NP. Why this theory with Goldilocks like power, able to solve problems not so difficult so as to rearrange our theory of tractible computation, but at the same time able to solve problems widely thought to be intractable on a classical computer?

Of course, you will object that I am asking too much of an interpretation. The interpretation is only supposed to make you feel good at night, when you crawl into bed with your copy of Cohen-Tannoudji et al, not to actually be *useful* (sorry, another low blow.) But I believe that an interepretation of quantum theory, which is obsensibly about resolving our conflicting feelings about the classical world we think we know and the quantum world, will only satisfy me if it comes along with an equal solution to resolving the conflicting feelings about why quantum computers are of the intermediate power we widely suspect them to be. Maybe, indeed this also offers an explanation for why there is little agreement over interpretations: the problem is related to a problem in computational compelxity, BQP?=BPP, whose resolution would represent a major insight in long standing difficult problems in computational complexity.

Dave you must reject your worship of false Bohmian Idols; verily recall the words of the Prophet:

“The way the quantum computer works is: the universe differentiates itself into multiple universes and each one performs a different sub-computation. The number of sub-computations is vastly more than the number of atoms in the visible universe. Then they pool their results to get the answer. Anyone who denies the existence of parallel universes has to explain how the factorisation process works.”

Praise be unto him. Amen.

Dave, excellent post. That exact same feeling is what gave me the impetus to go into quantum computation. I have had trouble explaining it to others and this post just laid it all out.

Great post, Dave. You’re absolutely right. Computation and number theory should just fall out of the right interpretation, in terms of higher topos theory. Connes et al have an inkling of this in their approach to the SM via Kreimer’s Hopf algebras, but they completely fail to take on board the possibilities of non-standard analysis.

“…even when the particles involved transverse regions where the vector potential vanishes.”

It is not the vector potential that vanishes, it is the magnetic field. Therefore, all your arguments must be invalid 🙂

Doh! Fixed. Thanks Kuas.

I’m flabbergasted, Dave. It’s as if you read my innermost thoughts, as if you sang from the mountaintops that which I always knew but could never express.

I now wish to bestow upon you the highest compliment of which I can conceive. From this day forward, you, David Morris Bacon, are no longer a physicist.

You are now a complexity theorist.Welcome to the tribe.Heh, Scott! I don’t see why he can’t be a Physicist as well.

Hehe, Scott makes a funny.

I think I’m a fan of the “shut-up and calculate” interpretation with a bit of added fancy computer science terminology.

Dave, I agree with some of the sentiments of your post. To me the idea that we can cleanly separate quantum theory into those parts which have a well defined operational meaning and “interpretation” is beginning to look increasingly ridiculous and is not particularly helpful for future progress. The distinction once served a reasonable purpose, which was to ring-fence the well understood parts of quantum theory, relevant to predicting the results of experiment, from the seemingly worrying implications of the theory. Physicists needed to focus on the former part of the theory in order to make the rapid progress that they did, and the time was not right for a detailed foundational analysis.

However, these days we have quantum information and computation, which do make use of the weirder aspects of quantum theory. Both this and the difficulties that we presently have with quantum gravity suggest to me that the time is ripe for a redefinition of what foundational studies should be about.

The unfortunate word “interpretation” suggests that foundational work should never impact the ring-fenced portion of the theory and should be confined to something like “finding the right language” for talking about quantum theory in order to avoid the difficulties with measurement, etc. Indeed, many philosophers actually like this distinction, because it makes “interpretation” a purely philosophical question that they can go off and study on their own, safe in the knowledge that they won’t find themselves proved wrong by physicists.

To me, this is a very bizzare thing to want to do, and I prefer the goal of setting a clear path for the future progess of physics, and maybe making some moves along the path as well. In fact, I find that most “interpretations” of any merit are not actually mere “interpretations” at all, since they do at least suggest different potential generalizations or refinements of the theory.

I realize that this is just a bit of a rant so far, and I need to justify some of the points made above, so more on this over at Quantum Quandaries in the near future.

Crap if Scott agrees with me I’m in trouble!

Here is an interesting spin-off from the BQP?=BPP obstruction in interpretations. BQP?=BPP is essentially an argument about the time complexity of these two classes. Therefore resolutions of the interpretation problems in quantum theory have to do, essentially, with time. Not with space, i.e. adhoc non-local theories, but with time. Further since PSPACE=BPP^{time travel} it can’t be something as drastic as closed time like curves, but has to be something more subtle and less powerful than a time machine. 😉

Hmmm (or is it hmm?). I generally hate labels and, in some small sense, that’s all an interpretation is. Tom Moore perhaps described it best by pointing out that classical physics consists of the successive layers: physical phenomona; conceptual explanation; mathematical description, whereas quantum physics is missing the conceptual layer in-between the phenomena and the mathematics. At least it is missing a single, consistent conceptual layer. Rather it has many.

Despite that fact that I don’t have the clout in the quantum community to do this, I’m going to go out on a limb and say that one of the problems is that many results we take for granted, particularly in regard to certain entanglement experiments, are

based on the interpretation, i.e. whether or not we are actually witnessing entanglement is dependent upon which interpretation we are using. In fact I would conjecture that this is why there stillisa plethora of interpretation, not to mention a host (albeit largely quiet) of skeptical physicists (most, though not all, outside our research area).Interestingly enough, efforts to convince a colleague of mine that true entanglement has actually been observed (personally, Aspect’s work is pretty convincing to me – no dependence on interpretation as far as I can tell) have highlighted the fact that correlation and entanglement are really two different things. Entanglement is actually the combination of correlation

andnon-local behavior. Some “entanglement” experiments seem to prove correlation but, without the Copenhagen interpretation being taken for granted,don’tdemonstrate non-locality. In fact, we are even designing our instruments and experiments under theassumptionthat the Copenhagen interpretation (or something similar) is true (i.e. measurements disturb processes), etc.).That being said, let me switch gears and say that quantum computation and information I believe is leading to a new interpretation that could bridge the gap between Copenhagen and classical assumptions. I’m rapidly becoming convinced that an explanation (for that is what an interpretation is) lies in statistical mechanics (note the work of the Horodeckis, Oppenheim, Cerf & Adami, and others). Perhaps most intriguingly (to me anyway) is a paper by O’Hara from a few years ago that hints at a link with rotational invariance and spin-statistics. While there are definite issues with O’Hara’s results, it’s getting closer to a realistic, testable explanation than anything else I’ve seen – one where tests are independent of the interpretation itself. Now, if those field theorists ever find a testable theory with more than 3+1 dimensions, I would bet that non-local behavior in our 3+1 dimensions is local in some other dimension(s).

At least for now, though, this is all just some vague intuition I have. It’s like having a few puzzle pieces that appear to be from the same puzzle but only by finding other pieces will we know for sure. Nonetheless, it seems to fit everything neatly together (at least in my head).

Of course, we could all take mick’s “shut up and calculate” approach (which was pioneered by Dirac, of course, with excellent results).

I may expound more on this at Quantum Moxie at some point.

Ian: The

whole pointof the Bell inequality is to demonstrate entanglement — i.e., to separate it from classical correlation — without assuming any interpretation. Admittedly, even many people who have studied quantum mechanics for years fail to understand this elementary point. I blame it on the way QM is usually taught (incorporating none of the insights of the last few decades).I can’t help but agree with Dave too. I can just add a few comments about how I would say it.

Really an interpretation is not a predictive theory at all, but a kind of

explanation. Bohmian “mechanics” is a good example. It doesn’t predict anything; the only question is what it might teach you. What it teaches you is that if you treat the tensor structure for joint states (or what physicists call “locality”) as peripheral to quantum mechanics, then you can embed quantum mechanics into a deterministic universe.This is valid, minor mathematical point, but it is no more than that. Of course locality is fundamental! The combination of the Bohmian gimmick and the Bell inequalities already prove that. It is also true that quantum computing proves it 1,000 times over. Bohmian mechanics cannot possibly be a useful explanation most of the time. You could equally well apply Bohm’s approach to classical probability, and interpret probability distributions as deterministic clouds of fluff; and it would be equally wrong-minded.

In the end, the explanation that I like best comes from operator algebras. Quantum mechanics is, or is based on, non-commutative probability, the natural non-commutative generalization of standard probability. This is not an explanation in the sense of a scientific etiology. Rather, it is a good pedagogical, mathematical explanation. It is also a mathematical fact (by Bell’s inequalities) that non-commutative probability cannot be placed inside commutative probability. That’s just life.

It is not clear that quantum probability needs an etiology. Not everything does; we don’t want to be like toddlers who ask “why” endlessly without ever moving onto new questions.

Scott: Absolutely, and that is entirely clear, but it wasn’t my point (and I agree with your comment about how QM is taught). Though Bell’s inequalities are not wholly classical – they still contain a quantum assumption (that is: correlation via the Pauli principle). Regardless, my point was about experimental verification of entanglement (which, admittedly, centers on Bell’s inequalities). In

someinstances it seems we get caught in orthodoxy and forget just what it is we are trying to prove. Aspect’s experiment very clearly demonstrates both correlation and non-locality (and thus entanglement) – that’s obvious – without making any prior assumption. My point is this: what if we didn’t know Bell’s inequalities existed? Could we “discover” entanglement purely via experiment? Aspect’s experiment answers this question with a resounding “yes,” but not all “entanglement” experiments do – that’s my problem. A good experiment should be interpretationally independent of theory.From Scottâ€™s reply to Ian I would infer that Scott would say one â€œmustâ€ accept non-locality, just as one â€œmustâ€ accept randomness or non-determinism, if one accepts standard quantum mechanics. Of course, some still seem to say one can choose to believe in â€œnonrealismâ€ or â€œcontextuallityâ€ and still retain a belief in locality. I have come to think that the necessary solution involves degrees of non-locality. Thus one â€œmustâ€ believe that correlations are non-local, but one â€œmustâ€ also believe that signals are local. (I think something like this is widely understood and accepted.)

I think the real headbuster is the realization that a theory without nonlocal signalling can have nonlocal correlations and still be mathematically consistent.

This gives us a three level locality hierarchy:

Bottom level: (most local) eg special theory of relativity. No nonlocal signalling and no non local correlations

Middle level: quantum mechanics non local correlations but no nonlocal signalling.

Top level: (most nonlocal) Newtonian gravity with instantaneous nonlocal signalling and correlations.

Now add the Popescu-Rohrlich nonlocal box and you have a fourth level of nonlocality above the middle level but below the top level.

I would like to speculate that this hierarchy could be extended to a continuum, perhaps indexed by a different Tsirelson bound at each level.

In googling around, I find that Barrett and Pironio have published a result which implies a level above the Popescu Rohrlich level based on cluster states. http://authors.library.caltech.edu/1833/01/BARprl05.pdf

An even wilder speculation: perhaps backward-in-time signalling can be proved to be even more nonlocal than instantaneous signalling thus giving us a new level above the previous top level.

The big question: Can these levels be rigorously well defined.?

James: Well, on the one hand, that additional level of nonlocality seems intuitively obvious, but on the other hand, since QM and even much of classical physics is probabilistic, I doubt that is a realistic physical expectation (this goes back to a point I made many months ago somewhere on this blog, that time – or rather the arrow of time – is a purely statistical/probabilistic phenomenon). I would hesitate to put Newtonian gravity in there as a case of non-locality since it’s not a fundamental theory, i.e. it’s a special case (approximation) of GR.

I think I just fell into my own trap so let me restate that last comment: obviously if there is nonlocality on the level middle/top levels as Jim points out, there’s nothing that precludes there being another level. From a probabilistic standpoint, it becomes increasingly less likely, but may still not be precisely zero, i.e. it might be like an extremely metastable state (somewhat like hydrogen spin transitions or, more extremely, proton decay).

Greg,

I agree with you that regarding QM as a generalization of classical probability theory gives the best intuition into what is going on, at least as far as the theorems of quantum information are concerned. The fact that such a good aid to intuition doesn’t gel with any of the main “interpretations” is a good reason to be sceptical of them. If nothing else, a good interpretation ought to act as the best guide to intuition that we have available.

However, this is not to say that I think this point of view actually resolves the conceptual difficulties that we have with quantum theory. At best, it provides a clue because we should expect a “good” interpretation to be consistent with the intuition it suggests. In my view, the basic question is what I call the “ontology problem”, i.e. exactly what can be said to be going on in reality for a quantum system in the absence of our interventions with it. This is my replacement for the “measurement problem”, since in the measurement problem you have to assume the reality of the wavefunction in order to even get the argument off the ground, which just begs the question.

Now, you may regard the “ontology problem” as a pure philosophy problem, to which physics can never supply a unique answer. You are free to do so if you wish, and your view would be supported by the interpretational debate to date. However, I tend to believe that a good solution to the problem, if it can be found, will lead to new physics, so I don’t think its a waste of time to study it.

Amen, Matt, Amen.

I agree with you that regarding QM as a generalization of classical probability theory gives the best intuition into what is going on, at least as far as the theorems of quantum information are concerned.It’s also perfectly adequate as far as existing physics is concerned, both theory and experiment. If it’s adequate for both math and physics purposes, is there really anything missing?

However, this is not to say that I think this point of view actually resolves the conceptual difficulties that we have with quantum theory.In my view, calling it a conceptual difficulty gives it the right context. Namely, it’s a psychology question and an education question.

You should know that most mathematicians have the same conceptual resistance to quantum probability as most other people. In some respects, they are even more stubborn, since they have a developed but entrenched understanding of deterministic dynamical systems and classical probability. The major exception, other than mathematical physicists, is operator algebraists. They are uniquely able to believe quantum mechanics, because it is consistent with the artificial ontology that they learn anyway.

Almost all mathematicians embrace some kind of artificial ontology of one kind or another — geometric topologists imagine 7-manifolds as if they are actually real, and so on. Operator algebraists have the good luck that their mathematical experience makes certain physics not only easier to learn, but also easier to believe.

I see this as positive evidence that good intuition, coming from good education, can make the “ontology” problem disappear. At the very least, it diminishes it.

Now, you may regard the â€œontology problemâ€ as a pure philosophy problem, to which physics can never supply a unique answer.Calling a question philosophical is sometimes about the same for me as calling it moot. I agree that physics

answersoften have philosophical strength. In that sense, the philosophical aesthetic is valuable. But if you call aquestionphilosophical, then you may be saying that if you don’t know if there really is a question.I tend to believe that a good solution to the problem, if it can be found, will lead to new physics, so I donâ€™t think its a waste of time to study it.I certainly agree that quantum information theory is very much worth studying. In one sense, it vindicates the philosophical bemusement of quantum probability. It shows that quantum probability

isa non-trivial topic in and of itself, and not just an unchanging cornerstone of physics. But I also think of it as an antidote to quantum philosophy. If people expected some other structureunderneathquantum probability, what turned out to be interesting instead is a mathematical theoryaboveit.Now, it is always possible that there is some unexpected, interesting new thing undernearth quantum probability. If you find that to be a useful motivation, that’s fine with me. But it would be better to admit that we don’t know that any such structure exists. It is not like an unsolved murder, where we can clearly see that an explanation is missing. Our philosophical angst with quantum probability isn’t good evidence that it is incomplete.

http://www.arxiv.org/PS_cache/quant-ph/pdf/0612/0612147.pdf

Title: Steering, Entanglement, Nonlocality, and the EPR Paradox

Authors: H. M. Wiseman, S. J. Jones, A. C. Doherty

This paper defines one more distinguishable level.

Evidence for my speculation that there is a continuous hierarchy of nonlocality?

I think so.

Comments?

If you are interested in details of how and why Bohm’s career in US was ruined, there is an excellent nuke-history book from Gregg Herken “Brotherhood of the Bomb”.

Bohm’s closest student buddy Joe Weinberg has volunteered to spy for soviets right at the start of the program and he gave quite detailed technical outlines of the secret project to a communist party operative. He did not get chance to become the second Fuchss – FBI people overheard this conversation with help of illegally-placed bug. Groves held emergency meeting in Washington about the security breach, there was massive increase in counter-spy effort and it became clear that Soviet consulate in San Francisco was eagerly trying to recruit people from the program. (Oppie even admitted that he was approached about this by somebody sent by russians). The fallout was that Weinberg did not go to Los Alamos as planed and his buddies Lomanitz, Bohm and Friedman (all commies) became very suspect – they were prevented from doing any work for government and soon were pushed out from the university as a security risk.

As an addendum, at Berkeley they used to keep a record of all of their physics graduate students (I believe the original records were kept by LeConte, but my memory could be faulty.) If you go to the Berkely physics library and look up David Bohm it has a comment about how he was an incredible student, scoring the highest score on the qualifying exam, but that they currently didn’t know where he was!

I would have thought that Bohmian mechanics was the ideal context in which to explore this question, since there is no handwaving about mechanism: every “quantum” behavior has a detailed explanation. But no-one has done the work yet (see how many hits “Bohmian computing” gets at Google).

I wonder if you ever were a Bohmian? Nature’s non local nature is not only acknowledged but insisted on. Config space makes room to decide not the possibilities but rather the certainties. The holistic nature of the wave gives one access to the Aleph and the particles displays your the results. What more could any Q-Bit flipper want?

Believe me when I say I was once a Bohmian.

But Aleph? Holistic? That’s Krishnamurti’s Bohm, not a physicist’s Bohm 🙂

I was referring to Georg Cantor’s Aleph as it relates to cardinality in set theory and how holistic in the mathematical sense could be considered such as in uncountable infinities as opposed to countable ones. Did you truly think I was speaking of some witch doctor? I’m sorry that particular witch doctor told Bohm enlightenment came when one didn’t (think that is). 😉