Tag Archives: papers

New Papers

I don’t normally like to just list new papers without commenting on them, but I don’t have much reading time at the moment so here are two that look interesting.

Firstly, Andrew Steane has a new paper entitled “Context, spacetime loops, and the interpretation of quantum mechanics”, which was written for the Ghirardi festschrift. Steane is best known for his work on quantum error correction, fault tolerance and ion trap quantum computing, which may not engender a lot of confidence in his foundational speculations. However, the abstract looks interesting and the final sentence: “A single universe undergoing non-unitary evolution is a viable interpretation.” would seem to fit with my “Church of the smaller Hilbert space” point of view. Steane has also addressed foundational issues before in his paper “A quantum computer only needs one universe”, and I like the title even if I am not familiar with the contents. Both of these are on my reading list, so expect further comments in the coming weeks.

The second paper is a survey entitled “Philosophical Aspects of Quantum Information Theory” by Chris Timpson. The abstract makes it seem like it would be a good starting point for philosophers interested in the subject. Timpson is one of the most careful analysers of quantum information on the philosophy side of things, so it should be an interesting read.

Quantum foundations before WWII

The Shtetl Optimizer informs me that there has not been enough contemplation of Quantum Quandaries for his taste recently. Since there has not been a lot of interesting foundational news, the only sensible thing to do is to employ the usual blogger’s trick of cut, paste, link and plagiarize other blogs for ideas.

Scott recently posted a list of papers on quantum computation that a computer science student should read in order to prepare themselves for research in quantum complexity. Now, so far, nobody has asked me for a list of essential readings in the Foundations of Quantum Theory, which is incredibly surprising given the vast numbers of eager grad students who are entering the subject these days. In a way, I am quite glad about this, since there is no equivalent of “Mike and Ike” to point them towards. We are still waiting for a balanced textbook that gives each interpretation a fair hearing to appear. For now, we are stuck trawling the voluminous literature that has appeared on the subject since QM cohered into its present form in the 1920’s. Still, it might be useful to compile a list of essential readings that any foundational researcher worth their salt should have read.

Since this list is bound to be several pages long, today we will stick to those papers written before the outbreak of WWII, when physicists switched from debating foundational questions to the more nefarious applications of their subject. This is not enough to get you up to the cutting edge of modern research, so more specialized lists on particular topics will be compiled when I get around to it. I have tried to focus on texts that are still relevant to the debates going on today, so many papers that were important in their time but fairly uncontroversial today, such as Born’s introduction of the probability rule, have been omitted. Still, it is likely that I have missed something important, so feel free to add your favourites in the comments with the proviso that it must have been published before WWII.

  • P.A.M. Dirac, The Principles of Quantum Mechanics, Oxford University Press (1930).
  • J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press (1955). This is the first English translation, but I believe the original German version was published prior to WWII.
  • W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift für Physik, 43, 172-198 (1927). The original uncertainty principle paper.
  • A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935).
  • N. Bohr, Can quantum-mechanical description of physical reality be considered complete?, Phys. Rev. 48, 696 (1935).
  • N. Bohr, The Philosophical Writings of Niels Bohr (vols. I and II), Oxbow Press (1987). It is a brave soul who can take this much Bohrdom in one sitting. All papers in vol. I and about half of vol. II were written prior to WWII. There is also a vol. III, but that contains post 1958 papers.
  • E. Schrödinger, Discussion of probability relations between separated systems, Proceedings of the Cambridge Philosophical Society. 31, 555-562 (1935).
  • E. Schrödinger, Die Gegenwärtige Situation in der Quantenmechanik, Die Naturwissenschaften. 23, 807-812; 824-828; 844-849 (1935). Translated here.
  • Birkhoff, G., and von Neumann, J., The Logic of Quantum Mechanics, Annals of Mathematics 37, 823-843 (1936).

Many of the important papers are translated and reproduced in:

  • J. A. Wheeler and W.H. Zurek (eds.), Quantum Theory and Measurement, Princeton University Press (1983).

Somewhat bizzarely it is out of print, but you should find a copy in your local university library.

I am also informed that Anthony Valentini and Guido Bacciagaluppi have recently finished translating the proceedings of the 5th Solvay conference (1927), which is famous for the Bohr-Einstein debates, and produced one of the most well-known photos in physics. It should be worth a read when it comes out. A short video showing many of the major players at the 1927 Solvay conference is available here.

Update: A draft of the Valentini & Bacciagaluppi book has just appeared here.

Anyone for frequentist fudge?

Having just returned from several evenings of Bayesian discussion in Vaxjo, I was inspired to read Facts, Values and Quanta by Marcus Appleby. Whilst not endorsing a completely subjectivist view of probability, the paper is an appropriate remedy for anyone who thinks that the frequentist view is the way to understand probability in physics, and particularly in quantum theory.

In fact, Appleby's paper provides good preparation for tackling a recent paper by Buniy, Hsu and Zee, pointed out by the Quantum Pontiff. The problem they address is how to derive the Born rule within the many-worlds interpretation, or simply from the eigenvalue-eigenstate (EE) link. The EE link says that if you have a system in an eigenstate of some operator, then the system posesses a definite value (the corresponding eigenvalue) for the associated physical quantity with certainty. Note that this is much weaker than the Born rule, since it does not say anything about the probabilities for observables that the system is not in an eigenstate of.

An argument dating back to Everett, but also discussed by Graham, Hartle and Farhi, Goldstone and Gutmann, runs as follows. Suppose you have a long sequence of identically prepared systems in a product state:

|psi>|psi>|psi>…|psi>

For the sake of definiteness, suppose these are qubits. Now suppose we are interested in some observable, with an eigenbasis given by |0>,|1>. We can construct a sequence of relative frequency operators, the first few of which are:

F1 = |1><1|

F2 = 1/2(|01><01| + |10><10|) + 1|11><11|

F3 = 1/3(|001><001| + |010><010| + |100><100|) + 2/3( |011><011| + |101><101| + |110><110|) + 1|111><111|

It is straightforward to show that in the limit of infinite copies, the state |psi>|psi>|psi>…|psi> becomes an eigenstate of Fn with eigenvalue |<psi|1>|^2. Thus, in this limit, the infinite system posesses a definite value for the relative frequency operator, given by the Born probability rule. The argument is also relevant for many worlds, since one can show that if the |0> vs. |1> measurement is repeated on the state |psi>|psi>|psi>…|psi> then there will be norm squared of the worlds where non Born-rule relative frequencies were found will tend to zero.

Of course, there are many possible objections to this argument (see Caves and Shack for a rebuttal of the Farhi, Goldstone, Gutmann version). One is that there are no infinite sequences available in the real world. For finite but large sequences, one can show that although the norm squared of the worlds with non Born probabilities is small, there are actually still far more of them than worlds which do have Born probabilities. Therefore, since we have no a priori reason to assign worlds with small amplitudes a small probability (which we do not because that is what we are trying to derive), we should expect to see non Born rule probabilities.

Buniy, Hsu and Zee point out that this problem can be avoided if we assume that the state space is fundamentally discrete, i.e. if |<phi|psi>| < epsilon for some small epsilon then |psi> and |phi> are actually the same physical state. They provide a way of discretizing the Hilbert space such that the small amplitude worlds dissapear for some large but finite number of copies of the state. They also argue that this discreteness of the state space might be derived from some future theory of quantum gravity.

I have to say that I do not buy their argument at all. For one thing, I hope that the conceptual problems of quantum theory have good answers independently of anything to do with quantum gravity. In any case, the question of whether the successful theory will really entail a discrete state space is still open to doubt. More seriously, it should be realized that the problem they are trying to solve is not unique to quantum mechanics. The same issue exists if one trys to give a frequentist account of classical probability based on large but finite ensembles. In that case, their solution would amount to the procustean method of just throwing away probabilities that are smaller than some epsilon. Hopefully, this already seems like a silly thing to do, but if you still have doubts then you can find persuasive arguments against this approach in the Appleby paper.

For me, the bottom line is that the problem being addressed has nothing to do with quantum theory, but is based on an erroneous frequentist notion of probability. Better to throw out frequentism and use something more sensible, i.e. Bayesian. Even then, the notion of probability in many-worlds remains problematic, but I think that Wallace has given the closest we are likely to get to a derivation of the Born rule for many-worlds along Bayesian lines.

Shameless self-promotion

As is traditional with physics blogs, it is time to indulge in a spot of shameless self-promotion of my own work. I have just posted a paper on quantum dynamics as an analog of conditional probability on the arXiv. This is about a generalization of the isomorphism between bipartite quantum states and completely positive maps, that is often used in quantum information. The main point is that it provides a good quantum analog of conditional probability, so it may be of interest to foundations-types who like to think of quantum theory as a generalization of classical probability theory.

The paper was completed in somewhat of a hurry, to get it out in time for the conference on Foundations of Probability and Physics in Vaxjo taking place this week, where I am due to give a talk on the subject. No doubt it still contains a few typos, so you can expect it to get updated in the next couple of weeks. Any comments would be appreciated.

More on the Vaxjo meeting to follow soon.

Realists on the counter attack

Martin Daumer, Detlef Duerr, Sheldon Goldstein, Tim Maudlin, Roderich Tumulka and Nino Zanghi, a collection of scholars noted for their advocacy or realist interpretations of quantum mechanics, and Bohmian mechanics in particular, have posted an article on quant-ph that attacks the idea that quantum theory is “fundamentally about information”. The article is a response to a recent essay in Nature by Anton Zeilinger, and is mainly a criticism of his particular viewpoint.

Most of their argument is based on the fact that interpretations like Bohmian mechanics offer a clear counterexample to various claims, such as that QM shows nature is fundamentally indeterministic and that the Bell and Kochen-Specker no-go theorems rule out realism. I think this is all fair enough, and I agree that it is well worth taking the time to become familiar with the Bohm interpretation if one is at all interested in foundations. It is quite amazing how often it can be used as an example to clear up confusion and misunderstandings about what we can infer from QM. On the other hand, this is a far cry from saying that Bohmian mechanics should be taken seriously as a description of reality. There are several arguments against doing so, which would take too long to go into right now. Perhaps I will do so in another post when I have more free time.

In any case, Zeilinger’s Nature essay seems a rather easy target to me. It was a short article, and there was clearly not enough space for any detailed arguments. Whether or not you think that Zeilinger in fact has any compelling arguments, there are many other contemporary approaches that also claim QM is about “information” in some sense, and it would be good to see a more in depth response to all of these from the realist camp. Examples include the quantum Bayesianism of Caves, Fuchs and Schack; the axiomatic approach of Bub, Clifton and Halvorson; and Hardy’s axiomatics.

Those of you who are waiting for Rovellifest 2 – fear not, for it is coming within the next week or so. For now, I feel like I need to write something on a topic I feel positive about, to aviod this blog descending into a sea of negative criticisms.

Rovellifest 1

Carlo Rovelli has recently put 3 papers on the arXiv, which have attracted some attention within the blogsphere (see here, here, here and here). The one that concerns us here at QQ is the paper about EPR in the relational approach to QM. I don't want to comment on the particular argument in that paper, which seems fine as far as it goes, but I do want to say a couple of things about Rovelli's approach in general, since it seems to be a popular topic at the moment. The main ideas of the approach can be found in Rovelli's original paper.

Here is an (admittedly cartoonish) summary:

1. We should shift attention from things like the measurement problem and instead try to derive QM from the idea that it is a theory of the information about one system that is available relative to other systems.

2. Quantum states are not absolute concepts and the state of a system is only defined relative to some other reference systems. Different reference systems do not have to agree on this state. If they do come to agreement it is only after the reference systems themselves interact with each other according to some Hamiltonian.

3. The question of whether a system has some particular property has no absolute meaning. However, some property of a system can be well-defined relative to some other system, provided the systems happen to have interacted in such a way that the second system records the appropriate information about the first system.

4. All the relational states just represent the subjective point of view that one system has about another. There is no absolute meaning to such states and no meaningful "wave-vector of the universe" can be constructed because there is no external system for it to enter into relations with.

5. This is all just a twist on the usual kind of relationalism that we have in other physical theories, e.g. special and general relativity.

In my opinion, there is a good deal wrong with relational QM as formulated by Rovelli, although I am not particularly opposed to relationalism in general. In this post, I'll make some comments about 4 and 5. A forthcoming "Rovellifest 2" post will point out a problem with 3, which I believe is more serious.

To address 5, it is worth noting a striking disanalogy between relational QM and other sorts of relational theories in physics. For example, in Newtonian mechanics we are very used to the idea that that there is no absolute meaning of the position of a particle A, but you can define its distance to a reference system B. This is generally different from the distance of A relative to another reference system C. Similarly, there is no absolute notion of when two events are simultaneous in special relativity, but this is well defined relative to any inertial reference frame.

However, in these cases it is always possible to find some transformation that relates the descriptions relative to different reference frames, provided you know the relations between the frames themselves, e.g. the Lorentz transformations in special relativity.

Now consider a quantum system composed of a subsystem A and two observers B and C. Suppose both B and C separately interact with A, possibly measuring different observables on A. Relative to B, A is supposed to have some definite property after this interaction and similarly for C. However, you generally can't convert between B and C's description of the situation if you only know the state of B relative to C. You can if they happened to measure the same observable, but that's a very special case.

In fact, the only way to relaibly convert between different observers relative states of the same system is to know the entire "wave-vector of the universe", something that is meaningless for Rovelli due to 4.

So, it seems we are left with two options:

1. Add in a "state of the universe" so that one can reliably transform between different descriptions of the same subsystem.

2. Abandon the classical notion that one can reliably transform between different descriptions of the same system.

Adopting 1 would essentially entail accepting an Everettian/many-worlds type scenario, something that Rovelli is keen to distance himself from. Therefore, I conclude that he must accept 2.

Abandoning reliable transformations is not a completely absurd thing to do, but it is important to note that this is a departure from what we usually mean by the term "relational". I am still not entirely convinced that it is consistent, although I haven't managed to think up a scenario where it would cause a problem yet. My suspicion is that it might be attacked by a "Wigner's Enemy" type of argument of the sort that was levelled against Chris Fuchs' Bayesian approach by Amit Hagar, which seems much more relevant to the relational approach than to its original target.

N.B. "Wigner's Enemy" is a new name I just thought up for the argument.  I figure he must be an enemy rather than a friend because friends don't usually try to erase your memory. 

The Free Will Theorem

Michael Nielsen recently posted a comment by John Sidles about a preprint by Kochen and Conway that was posted on the quant-ph arXiv yesterday. It's called "The Free Will Theorem", which is certainly a provocative title. Here's my comment on the paper that I left on Mike's blog.

Hmm… I had a look at this paper. The title sounds a bit crackpot, but given the status of the authors I was willing to give it a chance.

First of all the name “Free Will Theorem” opens a whole can of worms, which we probably don’t want to get into. Suffice to say, what they actually prove is an “indeterminism theorem”, i.e. they use a Bell-type argument + a no-signalling requirement to prove that nature must be indeterministic. I have heard similar arguments before, in particular Y. Aharonov and D. Rohlich mention it in their book, although I’ve never seen it written down formally before.

To call this a “free will theorem” one has to get into the debates about whether free will is compatible with determinism and, if not, whether indeterminism even solves the problem. Most contemporary philosophers seem to answer yes and no respectively, so I don’t think this theorem has much to do with free will, although it would take a lot more space to go through the arguments for and against thoroughly.

However, what I did think was interesting about the paper was the “hexagon universe” toy-model that they introduced in the second half of the paper. Given the current interest in understanding aspects of QM via simpler toy theories, e.g. nonlocal boxes and Spekkens toy theory, this might be a useful addition to the canon. I haven’t managed to decipher all the details of this model yet, so I’ll have to defer judgement on that.