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Interesting attempt to axiomatize quantum mechanics
Posted:
Sep 1, 2011 8:34 PM


I have been reading two very interesting papers by the same authors,
G. Chiribella, G. M. D'Ariano, and P. Perinotti,
"Probabilistic theories with purification", Phys. Rev. A 81, 062348 (2010), arXiv:9008.1583, and
"Informational derivation of quantum theory", Phys. Rev. A 84, 012311 (2011), arXiv:1011.6451.
These attempt an axiomatization of finite dimensional quantum mechanics which avoids purely mathematical axioms without physical meaning, such as an assumption that "pure states" of a quantum system are represented by rays in a Hilbert space.
I am hoping to get in touch with others who may be interested in discussing the ideas of these authors. To that end, I will attempt a partial review indicating what intrigues me and what disturbs me about these papers. I will refer to the first paper as CDP10 and the second as CDP11.
These are basically papers in pure mathematics which are motivated by fundamental physics. They are both long and intricately complicated. CDP10 is 40 pages with 64 definitions, 34 lemmas, 30 theorems, and 50 corollaries. CDP11 is 39 pages with 12 definitions, 78 lemmas, 20 theorems, and 51 corollaries. I have read only a fraction of these papers in detail, but enough that I think I have a sense of their results, methods, and probable correctness. I am impressed with them and think that they will be important even if should turn out that some of their results are in error.
There is some overlap between the two papers, but not a great deal. I think that most readers will need to read CDP10 in order to understand CDP11, which constantly refers to CDP10 for needed results. Both papers are well written, but in unusual notations invented by the authors, and the notations are different for the two papers. I thought the CDP10 notation was quite successful, but the CDP11 notation less so. For example, CDP11 uses a thickened horizontal line to denote equality instead of the usual "=", without explicitly informing the readers of this. I found this really puzzling even after I had guessed its meaning. What's wrong with "=", which everybody understands, and why make the reader guess the meaning of unfamiliar symbols?
As a mathematician, I was sometimes disturbed by the mathematical vagueness of some of the definitions, but I think most of them could probably be reformulated in a rigorous way. This is groundbreaking work, which may be akin to early 19'th century mathematics before notions like "continuity" were fully understood on a rigorous basis.
Although the papers are basically pure mathematics, the definitions are neither standard mathematical ones nor presented as is customary in pure mathematics. Mathematicians will probably need some familiarity with standard quantum mechanics to follow the paper. Reading it may be similar to the experience a bright high school student might have trying to read a text on abstract linear algebra on his own. The abstractions are not likely to be meaningful to someone completely unfamiliar with what is being abstracted.
I think it is unfortunate that the authors did not publish this in a more mathematically oriented journal. The Physical Review journals are not known for careful refereeing, and they do publish a lot of badly incorrect material. Experience has taught me to view with initial skepticism just about everything in these journals, including the papers under review.
A purely mathematical work of this complexity would be accepted by the mathematical community only after years of careful study by experts; publication in a seriously refereed journal would only be a first step. In this case, even the first step has not been accomplished.
I have identified a few points in these papers which seem to me questionable, but I am mindful that I could be mistaken. Even if they are incorrect, it is not clear that they could not be repaired. Even if they cannot be repaired it is not clear that the main arguments of the papers would be affected. In a work of this complexity, it would be surprising not to find any substantial errors.
I have considered writing the authors about the possible errors, but hesitate because of my stilllimited understanding of this intricate work. I am hoping to find someone else who may have read enough of it to either put me straight about my misunderstandings or to give me confidence that the problems I see may indeed be essential.
I will be happy to discuss this with anyone, but "discussion" is a reciprocal process. I would expect that anyone who wants to discuss it would *first* have read it to at least a depth to understand the main definitions and to evaluate technical arguments in detail. In the past when I have sent out such requests, I have gotten responses from anonymous addresses, sometimes signed only with a first name, of the order of: "Tell me what you think is wrong with it, and I will read it tonight and let you know."
Stephen Parrott



