Tag Archives: quantum

Temperamental Physicists

Posted on 29 August, 2006 by | 2 comments

The other day I was surfing on Chris Fuchs’ website, looking for a journal reference to one of his papers. I noticed that there is a new addition to his compendia of emails about the foundations of quantum mechanics. Now, depending on your temperament, you either love or hate reading such things. Certainly, the ideas are not as carefully formulated as they would be in a scientific paper, and some might find this annoying. Also, if you get annoyed by reading things that you don’t agree with, and you are bound to find at least something you don’t agree with unless your name is Chris Fuchs, then this is not for you. On the other hand, it does give some insight into how a fellow thinker about foundations formulates, develops and changes his ideas, which is something that most scientists go to great lengths to hide from public view. It also gives you an idea of his influences, and it is the reactions of other people to the emails and the quotes from philosophers that I will never be motivated to read myself, that really bring the thing to life. Speaking of the latter, I liked this quotation from William James – if anyone knows the source then please let me know:

The history of philosophy is to a great extent that of a certain clash of human
temperaments. Undignified as such a treatment may seem to some of my colleagues,
I shall have to take account of this clash and explain a good many of the divergencies
of philosophies by it. Of whatever temperament a professional philosopher is, he tries,
when philosophizing, to sink the fact of his temperament. Temperament is no conventionally
recognized reason, so he urges impersonal reasons only for his conclusions. Yet
his temperament really gives him a stronger bias than any of his more strictly objective
premises. It loads the evidence for him one way or the other, making a more sentimental
or more hard-hearted view of the universe, just as this fact or that principle would. He
trusts his temperament. Wanting a universe that suits it, he believes in any representation
of the universe that does suit it. He feels men of opposite temper to be out of key
with the world’s character, and in his heart considers them incompetent and ‘not in it,’
in the philosophic business, even though they may far excel him in dialectical ability.
Yet in the forum he can make no claim, on the bare ground of his temperament,
to superior discernment or authority. There arises thus a certain insincerity in our
philosophic discussions: the potentest of all our premises is never mentioned. I am
sure it would contribute to clearness if in these lectures we should break this rule and
mention it, and I accordingly feel free to do so.

It seems to me that the quote would still make sense if one replaced “philosophy” by “theoretical physics”. Now, before I get accused of being on the loony left of postmodernism and sociology of science, let me clarify that I do believe that the role of experiment is a big difference between the two subjects, and it is capable of resolving issues much more conclusively than argument alone. However, each physicist has to choose what topic to work on and which ideas and methods are most likely to lead to success. Despite the successes of the great edifice of modern theoretical physics, there are still dozens of possible views on which aspects of it are the most important and on why it all works in the first place. The temperament of the physicist plays a large role in coloring her/his attitude to such issues. For example, something as simple as the level of respect for authority can play an enormous role. This is evident in the currently ongoing debate about the success/failure of string theory (which I don’t claim to have enough expertise to say anything sensible about by the way) and in differing attitudes to whether the foundations of quantum mechanics is an important subject for a physicist to understand, or a load of metaphysical hogwash.

It takes all types to make the subject go forward, and it would be a boring life if we all agreed exactly what needed doing and how to go about doin it. However, I do think that if we paid greater attention to the fact that there need not be entirely scientific reasoning behind our differing points of view, and that this is OK – even normal – then that would help to diffuse some of the great controversies of modern science.

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fqxi award announcements

Posted on 2 August, 2006 by | 2 comments

There has been quite a bit of discussion on physics blogs recently about the announcement of the Foundational Questions Institute grants a couple of days ago. The stated aim of the institute is:

To catalyze, support, and disseminate research on questions at the foundations of physics and cosmology, particularly new frontiers and innovative ideas integral to a deep understanding of reality but unlikely to be supported by conventional funding sources.

In the blogs and comments, some have praised the choice of grantees, whilst others have criticized it for being too conservative, a waste of time, or for not including grants for some particular foundational topics that they think are important. The connection of fqxi to the Templeton foundation has also been extensively debated. Being a recipient of a grant myself, I obviously think they made at least some good choices, and am looking forward to being able to do some foundational work without having to pretend it has any practical applications in quantum information.

For those who complained about the choice of topics, I would just say that they can only work with the proposals they actually receive, so if people want to change the range of topics that are supported then I think the best way to do so is to submit a strong proposal to the next call.  To other critics, I would say that the worth of fqxi should ultimately be judged by the quality of research that is produced, rather than any predjudices one might have about what makes good foundational research, and this will become clear over the next couple of years.

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More on criteria for interpretations

Posted on 6 July, 2006 by | 3 comments

Well, my “big list” has proved to be my most popular blog post to date, thanks in no small part to a mention over at Uncertain Principles and a n u mber of other blogs. I know when I’m on to a good thing, so let’s stick with the topic for one more post.

The big news is that we have the first response to the criteria from an interpreter of quantum theory over at koantum matters. I would love to see responses from advocates of other interpretations, not because I expect many surprises, but more because it would help me to improve the criteria. I’d like to know if interpreters interpret the criteria in the way I intended.

One of the reasons for engaging in a project like this is that I personally don’t find any of the contemporary interpretations all that compelling. Advocates are often fairly good at arguing their case, so it can be hard to express exactly why a given interpretation makes me uneasy. It is fairly clear that, rightly or wrongly, most of the physics community agrees with me on this, since otherwise there would not be such an emphasis on Copenhagen and Orthodox Dirac-von Neumann ideas in undergraduate quantum mechanics courses. Other interpretations are usually dealt with in one or two lectures at the end of a course, if they are mentioned at all.

In my opinion, the most likely way that the debate on interpretations can be closed is if one interpretation makes itself indespensible for understanding quantum theory. This could be because it leads to new physics, but alternatively it could just lead to a far better way of explaining the phenomena of quantum theory to both students and the general public.

A useful comparison here is to Einstein’s approach to special relativity. In fact, the postulates of quantum theory have been compared to Einstein’s postulates by a variety of authors (e.g. see here and here). Despite Einstein’s insights, the plain fact of the matter is that almost all of the predicitive content of special relativity is contained in the Lorentz transformations, and their extension to the Lorentz and Poincare groups. Especially when doing quantum field theory, special relativity is almost always reduced to just this in modern applications. We could then contemplate starting with a mathematical axiomatization of the Lorentz group and never bother to teach students about Einstein’s postulates at all. This is supposed to be analogous to the current situation in quantum theory, where we cannot derive the whole theory from postulates that are explicitly physical in nature, but are ultimately forced to thinking in terms of abstract Hilbert spaces and the like.

In my view, the main advantage of Einstein’s approach is that it leads directly to the main phenomena of the theory without having to posit the Lorentz transformations to begin with. For example, by considering Einstein’s train thought experiments, we can understand why there is length contraction, time dilation and relativity of simultanaeity directly from the postulates themselves. We would consider a student ill equipped to study relativity if these arguments were not understood before diving into the derivation of the Lorentz transofrmations. In my opinion, it is this that makes relativity more easily understandable than quantum theory.

Therefore, I would argue that to replace orthodoxy in the classroom, an interpretation will have to provide a direct route to some of the main phenomena of quantum theory, as well as facilitating an elegant route to the full mathematical formalism. If not, the interpretation is always likely to remain interesting to only a small band of specialists. Part of the aim of the criteria is to try and make interpreters think about these sort of issues, and that was in particular the point of the “principles” criterion.

Another aim, and perhaps the main one, is to try and move the debate about interpretations forward a little bit. Currently, interpretaions are usually understood as counterpoints to Copenhagen/Orthodoxy. That is, we first explain these ideas, then poke holes in them by discussing the measurement problem, and finally introduce a new interpretation that is supposed to fix the problem. However, we now know that Copenhagen/Orthodoxy is just a small corner in a large space of possibilities, and not necessarily the most convincing of the possibilities at that. Therefore, it seems silly to focus exclusively on these ideas as the starting point. However, once we recognise this, it becomes difficult to formulate the conceptual problems of quantum theory in an interpretationally neutral way, since the measurement problem cannot even be formulated precisely unless we have already taken some stand on the meaning of the wavefunction. Nevertheless, unease about interpretations persists, so the criteria are partly designed to give interpreters a hard time by identifying the weaknesses in their proposals in a more neutral way. This is problematic because there are a number of known issues that only seem to apply to particular interpretations, e.g. it would be nice if the criteria forced many-worlds advocates to discuss the basis problem and the meaning of probability, which may not have analogs in other interpretations. One way of doing this would be to introduce a series of if … then … clauses into the criteria, e.g. if you take an ontological view of the wavefunction then explain the Born rule. However, this is obviously very inelegant and it would be nicer to capture the problems with all interpretations in a short simple set of criteria that applies to every interpretation equally.

With this in mind, it should be clear that the current list is far from final, and I would welcome any ideas on how to improve it.

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Professional Jealousy

Posted on 28 June, 2006 by | 22 comments

As some of you know, my alter ego works on quantum information and computation (I’ll leave you to decide which of us is Clark Kent and which is Superman). My foundations personality sometimes feels a twinge of professional jealousy and I’ll tell you why.

In quantum computation we have a set of criteria for evaluating proposed experimental implementations, known as the diVincenzo criteria. These tell you what is required to implement the circuit model of quantum computation, and include things like the ability to prepare pure input states and the ability to perform a universal gate set. Of course, you might choose to implement an alternative model of computation, such as the measurement based models, and then a different set of criteria are applicable. Nevertheless, talks about proposed implementations often proceed by explaining how each of the criteria is to be met in turn. This makes it very clear what the weak and strong points of the implementation are, since there are usually one or two criteria that present a significant experimental challenge.

In contrast, there is no universally accepted set of criteria that an interpretation of quantum mechanics is supposed to meet. They are usually envisioned as attempts to solve the nefarious “measurement problem”, which is actually a catch-all term for a bunch of related difficulties to which different researchers attach different degrees of significance. The question of exactly what an interpretation is supposed to do also varies according to where one is planning to apply it. Is it supposed to explain the emergence of classical mechanics, help us understand why quantum computation works, give us some clues as to how to construct quantum gravity, or simply stand as a work of philosophical elegance?

It seems to me that the foundations community should have, by now, cracked their heads together and come up with a definitive list of issues on which an interpretation has to make a stand, before we are prepared to accept it as a viable contender. Then, instead of reading lots of lengthy papers and spending a lot of time trying to work out exactly where the wool has been pulled out from under our eyes, we can simply send each new interpreter a form to fill in and be done with it. Of course, this is bound to be slightly more subjective than the di Vincenzo criteria, but hopefully not by all that much. For what it’s worth here is my attempt at the big list.

The first six criteria would probably be agreed upon by most people who think seriously about foundations.

  • An interpretation should have a well-defined ontology.
    • To begin with, you need to tell me which things are supposed to correspond to the stuff that actually exists in reality. This can be some element of the quantum formalism, e.g. the state vector, something you have added to it, e.g. hidden variables, or something much more exotic, e.g. relations between things without any definite state for the things that are related, correlations without correlata etc. This is all fine at this stage, but of course the more exotic possibilities are going to get into trouble with the later criteria.
    • At this stage, I am even prepared to allow you to say that only detector clicks exist in reality, so long as you are clear about this and are prepared to face the later challenges.
    • As a side note, some people might want to add that the interpretation should explicitly state whether the quantum state vector is ontological, i.e. corresponds to something in reality, or epistemic, i.e. something more like a probability distribution. I am inclined to believe that if you have a clear ontology then it should also be clear what the answer to this question is without any need for further comment. I am also inclined to believe that this fixation on the role of the state vector is an artifact of taking the Schroedinger picture deadly seriously, and ignoring other formalisms in which it plays a lesser role. For instance, why don’t we ask whether operators or Wigner functions are ontological or epistemic instead?
  • An interpretation should not conflict with my direct everyday experience.
    • In everyday life, objects appear to be in one definite place and I have one unique conscious experience. If you have adopted a bizarre ontology, wherein this is not the case at the quantum level, you have to explain why it appears that it is the case to me. This is a particularly relevant question for relationalists, Everettistas and correlationalists of course. It is also not the same thing as…
  • An interpretation should explain how classical mechanics emerges from quantum theory.
    • Why do systems exist that appear to have states represented by points in phase space, evolving according to the classical evolution equations?
    • Note that it is not enough to give some phase space description. It must correspond to the description that we actually use to describe classical systems.
    • Some people might want to phrase this as “Why don’t we see macroscopic superpositions?”. I’m not quite sure what it would mean to “see” a macroscopic superposition, and I think that this is the more general issue in any case.
    • Similarly, you may be bothered by the fact that I haven’t mentioned the “collapse of the wavefunction” or the “reduction of the wavevector”. Your solution to that ought to be immediately apparent from combining your ontology with the answer to the present issue.
    • Some physicists seem to think that the whole question of interpretation can be boiled down to this one point, or that it is identical with the measurement problem. I hope you are convinced that this is not the case by now.
  • An interpretation should not conflict with any empirically established facts.
    • For example, I don’t mind if you believe that wavefunction collapse is a real physical process, but your theory should be compatible with all the systems that have been observed in superposition to date.
  • An interpretation should provide a clear explanation of how it applies to the “no-go” theorems of Bell and Kochen Specker.
    • A simple answer would be to explain in what sense your interpretation is nonlocal and contextual. If you claim locality or noncontextuality for your interpretation then you need to give a clear explanation of which other premises of the theorems are violated by your interpretation. They are theorems, so some premise must be violated.
  • An interpretation should be applicable to multiparticle systems in nonrelativistic quantum theory.
    • Some interpretations take the idea that the wavefunction is like a wave in real 3d space very seriously (the transactional interpretation comes to mind here). Often such ideas can only be worked out in detail for a single particle. However, the move to wavefunctions on multiparticle configuration space is very necessary and needs to be convincingly accomplished.

The next four criteria are things that I regard as important, but probably some people would not give them such great importance.

  • An interpretation should provide a clear explanation of the principles it stands upon.
    • For example, if you claim that your interpretation is minimal in some sense (as many-worlds and modal advocates often do) then you need to make clear what the minimality assumption is and derive the interpretation from it if possible.
    • If you claim that “quantum theory is about X” then a full derivation of quantum theory from axioms about the properties that X should satisfy would be nice. Examples of X might be nonstandard logics, complimentarity, or information.
  • No facticious sample spaces.
    • OK this is a bit of a personal bugbear of mine. Some interpretations introduce classical sample spaces (over hidden variable states for instance) or generalizations of the notion of a sample space (as in consistent histories). Quantum theory is then thought of as being a sort of probability theory over these spaces. Often, however, the “quantum states” on these sample spaces are a strict subset of the allowed measures on the sample space, and the question is why?
    • I allow the explanation to be dynamical, in analogy to statistical mechanics. There we tend to see equilibrium distributions even though many other distributions are possible. The dynamics ensures that “most” distributions tend to equilibrium ones. Of course, this gets into the thorny issues of the foundations of statistical mechanics, but provided you can do at least as good a job as is done there I am OK with it.
    • I also allow a principle explanation, e.g. some sort of fundamental uncertainty principle. However, unlike the standard uncertainty relations, you should actually be able to derive the set of allowed measures from the principle.
  • An interpretation should not be ambiguous about whether it is consistent with the scientific method.
    • Some interpretations seem to undermine the very method that was used to discover quantum theory in the first place. For example, we assumed that experiments really had outcomes and that it was OK to reason about the world using ordinary deductive logic. If you deny any of these things then you need to explain why it was valid to use the scientific method to arrive at the theory in the first place. How do you know that an even more radical revision of these concepts isn’t in order, perhaps one that could never be arrived at by empirical means?
  • An interpretation should take the great probability debate into account.
    • Quantum theory involves probabilities and some interpretations take a stand on the fundamental significance of these. Is the interpretation consistent with all the major schools of thought on the foundations of probability (propensities, frequentism and subjectivism), at least as far as these are themselves consistent? If not, you need to be clear on what notion of probability is actually needed and address the main arguments in the great probability debate. Good luck, because you could spend a whole career just doing this.

The final three criteria are not strictly required for me to take your interpretation seriously, but addressing them would score you extra bonus points.

  • An interpretation should be consistent with relativistic quantum field theory and the standard model.
    • Obviously, you need to be consistent with the most fundamental theories of physics that we have at the moment. However, the conceptual leap from nonrelativistic to relativistic physics is nontrivial and it has implications for ontology even if we forget about quantum theory. Therefore, it is OK to just focus on the nonrelativistic case when developing an interpretation. QFT might require significant changes to the ontology of your interpretation, and this is something that should be addressed eventually.
  • An interpretation should suggest experiments that might exhibit departures from quantum theory.
    • It’s good to have something which can be tested in the lab. Interpretations such as spontaneous collapse theories make predictions that depart from quantum theory and these should be investigated and tested.
    • However, even if your interpretation is entirely consistent with quantum theory, it might suggest novel ways in which the theory can be modified. We should be constantly on the lookout for such things and test them wherever possible.
  • An interpretation should address the phenomenology of quantum information theory.
    • This reflects my personal interests quite a bit, but I think it is a worthwhile thing to mention. Several quantum protocols, such as teleportation, suggest a strong analogy between quantum states (even pure ones) and probability distributions. If your interpretation makes light of this analogy, e.g. the state is treated ontologically, then it would be nice to have an explanation of why the analogy is so effective in deriving new results.

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Anyone for frequentist fudge?

Posted on 14 June, 2006 by | 4 comments

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.

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Vaxjo Meeting

Posted on 13 June, 2006 by | Leave a comment

I returned this weekend from the meeting on Foundations of Probability and Physics at the University of Vaxjo in Sweden. There were many interesting talks, so I'll just mention a few of them that I found particularly inspiring.

– Giacomo Mauro d'Ariano explained his axiomatization of quantum theory, inspired by observations from quantum state and process tomography. One of the nice features of this is that he gives an operational definition of the adjoint. Why the observables of QM should form an algebra from an operational point of view has been a topic of recent debate amongst foundational people here at Perimeter, so this could be a piece of the puzzle.

– Rüdiger Schack explained what it might mean for quantum randomness to be "truly random" from a Bayesian point of view, using the concept of "inside information" that he has developed with Carlton Caves.

– Philip Goyal gave another axiomatization of quantum theory. I'm not sure whether the framework he uses is that well-motivated (especially the sneaky way that complex numbers are introduced). On the other hand, one of his axioms has the flavor of an "epistemic constraint", which gels nicely with ideas that have been expressed earlier by Chris Fuchs and Rob Spekkens.

– Joseph Altepeter gave another excellent talk about the state of the art Bell inequality experiments currently going on in Paul Kwiat's group.

– John Smolin outlined speculative ideas that he and Jonathan Oppenheim have developed that applies the concept of locking quantum information to solve the black hole information loss problem. 

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Shameless self-promotion

Posted on 5 June, 2006 by | Leave a comment

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.

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Rovellifest 2

Posted on 16 May, 2006 by | 9 comments

I thought it was about time I got around to finishing my comments on Rovelli’s “Relational QM” programme.

Relational QM (RQM) has a lot in common with the Everett/many-worlds interpretation, so it should be no surprise that it shares some of the same difficulties. In my opinion, the “basis problem” also applies to RQM, and one cannot appeal to decoherence in order to solve it as one does in many-worlds. Before discussing this, let me summarize the main differences between RQM and many-worlds:

  • In Everett, the state-vector of the universe is the full description of reality. It always evolves unitarily, but different observers can have different subjective impressions of reality depending on how they are described within this state.
  • In RQM there is no state-vector of the universe. State-vectors always describe the point of view that one subsystem has about another system. State vectors are therefore always subjective descriptions of reality.
  • In Everett, the concept of measurement is an emergent phenomenon that applies when a macroscopic system interacts with a microscopic one. The theory of decoherence is used to explain why measurement results appear to be stable.
  • In RQM, Rovelli states explicitly that he doesn’t want to treat microscopic systems any differently from microscopic ones. For example, if two electrons interact, then it is valid to think that one of the electrons acts as a measuring device on the other and vice versa. One description is valid from the point of view of one of the electrons and the other is valid from the point of view of the other electron.

The appeal to decoherence in Everett is designed to address the “basis problem”, which arises due to the ambiguity over which baisis the states are decomposed in. For example, suppose two qubits start in the (unnormalized) state

(|0> + |1>)|0>

and interact so that they end up in the state

|00> + |11> .

This is a typical example of a “measurement” interaction and we might want to say that the second qubit has measured whether the first qubit is in the state |0> or |1>. In Rovelli’s formulation, the state of the first qubit is definitely either |0> or |1>, relative to the second qubit, with 50/50 probabilities of each being the case.

However, we could equally well write the final state as

(|0> + |1>)(|0> + |1>) + (|0> – |1>)(|0> – |1>).

If the qubits are actually spin-1/2 particles and |0> and |1> are spin up and spin down in the Z-direction, then this is a decomposition of the state in the spin-X basis. Therefore, we might equally well say that the second qubit has measured whether the first qubit is in the state (|0> + |1>) or (|0> – |1>). In Rovelli’s formulation, we ought to be able to say that the first qubit is either in the state (|0> + |1>) or the state (|0> – |1>) with 50/50 probabilities.

Note that, this is not only an issue with the particular state |00> + |11>. Any bipartite state can be decomposed according to any basis for one subsystem, although the relative states of the other system will not generally be orthogonal.

I have seen no discussion of this issue from Rovelli. He seems to assume that there just is some natural basis in which to do the decomposition. I think the possible solutions are:

  • Accept multiple descriptions: The state of one subsystem is not only relative to another subsystem, but it is also relative to an arbitrary basis choice. The problem with this is that it does not explain why our subjective experience is always according to one particular basis. I always feel like I am in one particular location, observing one particular thing, and I am incapable of regarding myself as being in a superposition of two locations, despite the fact that such a decomposition of the wavefunction almost certainly exists.
  • Stipulate a basis: For example, the position basis might be a natural choice, since it generically corresponds to our everyday subjective experience. The question then arises as to why this basis is chosen rather than some other. What is there within the formalism of QM that compells us to make this choice?
  • Appeal to decoherence: Decoherence theory usually supplies a “pointer basis” in which the results of measurement outcomes are almost exactly stable. This is the usual solution of the Everettians. However, if Rovelli takes this option then he has to back away from the position that microscopic systems are to be treated in exactly the same way as macroscopic one. It would no longer make sense to talk of a single electron acting as a measuring device.
  • Use the biorthogonal decomposition: Most bipartite states have a unique decomposition of the form \sum_j a_j | phi_j> |psi_j>, where <phi_j | phi_k> = \delta_{jk} and <psi_j | psi_k > = \delta_{jk}. We could simply stipulate that this basis is the correct one to do the decomposition in. This is the solution advocated in some variants of the modal interpretation. Problems include the fact that there are special states like the one above (admittedly they form a set of measure zero) for which the decomposition is not unique. Also, the biorthogonal basis does not always correspond exactly to our subjective experience, e.g. it may be close to, but not exactly equal to, the position basis.

    My impression is that none of these solutions would completely appeal to Rovelli, so it would be interesting to see what he says about the matter. However, if we combine this issue with the previous comments I made, then I have a hard time seeing how the Everettian/many-worlds ontology can really be avoided in this sort of approach.

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    Realists on the counter attack

    Posted on 25 April, 2006 by | 7 comments

    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.

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    Rovellifest 1

    Posted on 17 April, 2006 by | 10 comments

    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. 

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