What's the worst thing about the tomographic approach to QM?

Question

I saw a paper on arXiv that referenced this approach to an ontology of QM:

Phys.Lett. A213 (1996) 1, S. Mancini, V. I. Man'ko, P.Tombesi Symplectic tomography as classical approach to quantum systems
http://arxiv.org/abs/quant-ph/9603002

An introduction to the tomographic picture of quantum mechanics
http://arxiv.org/abs/0904.4439

Answer 1

The worst thing about it is that it is one of a large number of misguided papers that try to deny that the world is not and cannot be classical. The Wigner function may be calculated from a wave function or a density matrix and is approximately related to the probability distribution but not exactly.

The Wigner function is constrained and the precise probabilities that an observable will have a particular value have to be calculated as the expectation value of the corresponding projection operator (a linear operator on a Hilbert space) - and can never be fully encoded in a distribution function that is a function of several commuting variables simply because the real variables, as quantum mechanics and the experiments verifying it guarantees, do not commute.

There are several basic facts about quantum mechanics - the probabilistic character and the nonzero commutators of the observables (i.e. the uncertainty principle) are two important examples - and any paper denying those facts is simply wrong and doesn't deserve a further discussion.

To emphasize that they won't be convinced by any evidence, they motivate their paper - in the second sentence of this paper - by a "permanent wish to understand quantum mechanics in terms of classical probabilities." Sorry, this is not science. This wish has been ruled out and it can never be "unruled out" again. In science, hypotheses are never "permanent". Hypotheses are only acceptable as long as they haven't been falsified. Falsification ends the debate and it's the case of the hypothesis that the microscopic physics may be described by classical physics.

Answer 2

The answer by Lubos and this unfortunate sentence in page one of the older paper (as quoted by Lubos) "permanent wish to understand quantum mechanics in terms of classical probabilities" gives the impression that this Tomographic approach is trying to do something known to be impossible.

In fact the field has become the basis of a wide ranging series of practical techniques as discussed in the Wikipedia article on Quantum Tomography. Amongst its applications are quantum optics and practical quantum computation.

So I will just mention a few points of clarification here. Clearly any simplistic attempt to "reconstruct" $\Psi$ from knowledge of any (say) position (X) probability distribution cannot work. Even knowing the momentum (P) probability distribution as well is not always enough. So a price has to be paid to reformulate $\Psi$ in this kind of way. In summary one needs to know these probability distributions:

$T(\mu, \nu) = \mu X + \nu P$

As you can see this is a parameterised family with 2 dimensions worth of parameters.

Using Radon transforms, Wigner functions and the like the theory is able to reconstruct (well not $\Psi$) but the density matrix $\rho$.

Now there are still some issues here.

1. Q. How to actually obtain this data? A. By doing multiple measurements on similarly prepared quantum states. This is how Tomography has found applications in Optics, where there will be plenty of similar source photons to use.

2. Q. Are $\mu$ and $\nu$ not continuous, leading to error? A. Yes. So some form of statistical sampling logic needs to be added in practice.

3. Q. What about other observables apart from X and P? A. Yes the theory can be generalised to other classes of observables. This is one area where the theory may have current limits though.

4. Q. From a Hilbert space point of view what is going on? A. (I think) that what is happening is that alternative representations (to X, P, etc) are being constructed. One example they derive which is familiar to me is the coherent state representation. In this representation $\Psi$ becomes a function of Z=(X,P) as in Z = X + iP. Every coherent state is defined by a single (complex - hence 2D) point in this representation and the vacuum is 0 = origin. It is well known that the coherent states form an overcomplete basis for $\Psi$ and this is represented in their framework.

5.Q. Is Tomography really an Interpretation of QM? A. I dont think so, just a set of mathematical tools to be used in practical and maybe theoretical analysis of QM systems.

So what is the "worst thing"? - lack of a good coherent presentation of the entire area and its research philosophy (at least if those cited papers are anything to go by).

Answer 3

The second paper is available as Phys. Scr. 79 (2009) 065013 (29pp). An editor and a referee or two think there are not too many things wrong with it, or that it has enough good things about it. One assumes Luboš was not a referee. The whole community doesn't vote on whether a paper is accepted into a journal, of course, so a pinch of salt, but as of now there are 26 citations to the 2009 paper in published papers, according to Web of Science (which is not shabby at all). The editor required them to cut down the paper significantly, 64pp→29pp, so one of the worst;-) things about the arXiv version is mitigated in the published paper.

I take this kind of approach to QM to be on the edge of engineering and interpretation. There is an engineering problem that is addressed by tomography — how do we determine what quantum observables we are measuring (and the relationships between them, expressed in terms of the mathematical ansatz we use to model them) and what quantum state we have prepared, from the basic classical raw data points (that are written in lab note books or stored in computer memory) and whatever statistics of that raw data we may construct? Note that the engineering problem begins with classical information and ends with quantum observables and a quantum state, and, as such, is intimately concerned with the interpretation of QM, at least in the old-fashioned vaguely formulated terms of the Copenhagen and Bohr's interpretations. For more modern interpretations, which often layer some kind of metaphysics of the quantum state on top of the raw data, the relevance is not as clear-cut.

The worst thing about the second paper may not be something for which it can be hung out to dry, just because no one else has achieved definitude: it does not construct a compelling account. Specifically, it does not compel Luboš to take it seriously. This is an extraordinarily difficult balance to achieve, however it looks to me that if current progress is maintained someone will find an adequate combination of simplicity and mathematical and philosophical sophistication in the next 10 years.

Having written part of this before Roy Simpson's Answer, I have up-voted his Answer and commend it to other people as a counterpoint to the approach I have taken. I commend the papers cited in the Question as useful for engineering purposes and as a valiant attempt at interpretation.