June 2000, “References for Gleason”

Some relevant references. The important ones are Gleason's paper, Cooke et al.'s paper, and Pitowsky's paper.

1. R. Cooke, M. Keane, and W. Moran, “An Elementary Proof of Gleason's Theorem,” Math. Proc. Camb. Phil. Soc.98, 117–128 (1981).

2. A. M. Gleason, “Measures on the Closed Subspaces of a Hilbert Space,” J. Math. Mech.6, 885–894 (1957).

3. I. Pitowsky, “George Boole's ‘Conditions of Possible Experience’ and the Quantum Puzzle,” Brit. J. Phil. Sci.45, 95–125 (1994).

4. I. Pitowsky, “Infinite and Finite Gleason's Theorems and the Logic of Indeterminacy,” J. Math. Phys.39, 218–228 (1998).

5. F. Richman and D. Bridges, “A Constructive Proof of Gleason's Theorem,” J. Func. Anal.162, 287–312 (1999).

December 2000, “New Ideas and Old Men”

Thanks for the new ideas. Let me try to make a few comments. I'm in a smoky hotel bar right now; I wonder how this is affecting my laptop's health? I know it's wreaking havoc on mine. I'm still in Vienna, by the way. I leave Tuesday morning. Unfortunately I haven't gotten an ounce written of what I had hoped to.

Renesism 1:This got me thinking about this stuff, and it seems that in some sense what we're gathering information about, when we make measurements, is the ability to predict future measurements.

You haven't read my paper with Kurt Jacobs yet, have you? [See quant-ph/0009101.] Come on, at least read the Introduction and Footnote 27.

April 1996, to Michiel van Lambalgen, “Thesis”

About myself (in case you are wondering) I am a physicist. My research specialty is “quantum information theory,” a field that concerns a hodgepodge of things including quantum cryptography, quantum computing, and statistical inference problems having to do with quantum mechanical systems. You may have heard of some of these things through Paul Vitanyi or André Berthiaume (Vitanyi's postdoc). I am presently a postdoc working for Gilles Brassard and Claude Crépeau in Montréal; starting October, I will have a three year position at Caltech.

I was once interested in the mathematics of randomness because, though I am Bayesian through and through for all other uses of probability, I believed that probabilities for quantum mechanical measurement outcomes were something different…in fact something more akin to the frequentist conception. Thus I put a lot of effort into studying von Mises, Church, Kolmogorov, Martin-Löf, Chaitin, etc. (I didn't find your papers until I had pretty much abandoned this belief, though I'm not sure that I am completely over it!) I had hoped that there might be some mathematical connection between the structures found in quantum theory (vector spaces, positive-operator valued measures, etc.) and the structures required to formalize the notion of random sequences…at least that was my motivation.

30 August 1998, “Desiderata”

I just read over your four desiderata for an acceptable “über-quantum theory” for the third time. (When I finish this note, I'll read them over for the fourth time … at the very least for simile's sake.) They're not so uninspired! (… to use your words, that is.) But I do hope you will keep your promise of writing a longer exposition.

In the mean time, let me ask you for one point of clarification about the connection between your Desideratum #1 and my latest happy thoughts.

Kentism 1:1. A theory in which some elementary, and precisely defined, components constitute the basic statements from which propositions about an external and objective reality can be built. Maybe trajectories à la Bohm, maybe collapse events à la GRW, maybe events of more general form à la consistent historians, maybe the evolution of the classical gravitational field in some (unknown) theory that links classical metric and quantum matter. Maybe something quite different from any of the above. But, at any rate, something which allows us to talk about what happened in the past, in the absence of observers, and which allows us to resolve the measurement problem by replacing the vague notion of measurement by something precise.

03 April 2000, “Foundations in the Light of Information”

Below is a slightly modified version of the invitation letter I sent everyone. (The only change is an update on the attendance.) Looking forward to seeing you again. (BTW, I hope you'll take the problems list seriously.)

Dear friends,

Gilles Brassard and I are organizing a little meeting in Montréal from Wednesday May 17 through Friday May 19 on the subject of “Quantum Foundations in the Light of Quantum Information and Cryptography.” We hope you can come. Mainly the purpose of the meeting is to gather a group of our friends who think that some aspects of quantum foundations are a little more mysterious than they ought to be and, importantly, are intrigued by the idea of applying quantum information to the task of cleaning things up. The atmosphere of the meeting should be quite relaxed with plenty of time for discussion and/or private brooding: we only ask that you be prepared to give a 20 to 30 minute talk (though longer talks are welcome too) expressing your point of view and any avenues for progress you foresee. Attendants are also encouraged to compile and share a list of concrete problems whose solution would tell us something novel about the quantum foundations.

February 1996, “Goo-Losh”

Mmm, goulash…is that how you spell it? Thanks for the tip. I haven't heard the reaction of Bennett and company on this business yet, but when I do, I'll send a report your way.

I've been thinking about Kochen–Specker theorems lately; they're very nice you know…especially the ones with small numbers of vectors. Implications for “quantum information”? There must be! In particular, why can't one go much further in this effort to prove the theorem with smaller and smaller numbers of vectors? Why is there no proof of this property (i.e., no noncontextual hidden variables for the outcomes of standard von Neumann measurements), the very second you admit a measurement that is nonorthogonal to the others? Just food for late night thoughts.

March 1996, “The Commitments”

It's a good movie; did you ever see it?

Suppose you need some cash and so you go to your Automated Teller Machine (ATM). You insert your bank card and the machine asks you for your Personal Identification Number (PIN). It does this for the obvious reason that it wants to know that you are indeed who you say you are before it hands over the money. That's a good safeguard. However suppose, as really did happen a year or two ago, someone had set up a fake ATM in place of the real one.

18 November 1997, “Bolt from the Blue”

I saw Adrian Kent's four papers on the server early this morning, and I was reminded of Léon Rosenfeld's description of Bohr's reaction to the EPR paper:

I suspect the air in Montréal will be just as this when you all awake today. How I wish I were there to see the excitement!

I hope for the sake of our field, and a generally more exciting world, that Kent is right! As you all know, this sceptical physicist (i.e., me) was always a little wary of the Mayers–Lo–Chau strong claim of having considered all possible quantum protocols. I would like to hear the Montréal verdict once it is in.

A good day to all!

31 August 1999, “Subliminal Messages”

Quantum Foundations in the Light of Quantum Cryptography

Quantum Foundations in the Light of Quantum Cryptography

Quantum Foundations in the Light of Quantum Cryptography

Quantum Foundations in the Light of Quantum Cryptography

November 1997, “Martha White and Her Flour”

[Note: Words supplied by Greg Comer. See notes to Asher Peres, dated 17 October 1997, Greg Comer, dated 25 November 1997, and Carlton Caves, dated 21 December 1997.]

October 1998, “Oh Magic Eight Ball”

Democritus said, “All the world is but Atom and Void.” I've decided (with Kiki's help) that I much more like the metaphor, “All the world is but Magic Eight Balls and Questions.” See the resemblance? You couldn't have atom without void; but you could have void without atom. You couldn't have Questions without Magic Eight Balls; but you could have Magic Eight Balls without Questions. The sole remainder of archaic atomism is that “atoms” be “repositories.” Not repositories of “properties.” But repositories of answers to the questions we might ask. To keep these sentences from being contradictory, those answers can't exist before we ask them (else they be properties).

April 1999, “Thinking Kantian Thoughts?”

The other day I wrote the little essay below [cf. “Fuchsian Genesis” addressed to Greg Comer, dated 22 April 1999] for myself, and then searched up a reason to send it to my friend Greg Comer. It would have been better to send it to you I think. It strikes me as sounding a little Kantian, though with an extra twist: namely, that we are even further removed from the “ding an sich” than he had imagined. On my view, we know that the quantum formalism CANNOT correspond to reality (whatever it may be). My understanding of Kant is that he would have said that classical physics may or may not be a reflection of the real thing, but we can never know. It does form part of the categories of understanding and is therefore necessary to make sense of the world; but correspondence to reality is a different issue – in fact, one that cannot be decided. In the case of the quantum world, I think it is (reasonably) safe to assume that the terms in quantum theory cannot be mapped to any sort of “ding an sich.” That's why it strikes me as a still stronger form of Kantianism.

Am I way off track here? Any thoughts?

Jon's Reply

February 2000, “The Quantum View”

What a really nice article! I read it once this evening with dinner (at a place that served grilled chicken – not so easy to scroll through a computer screen under those conditions!). Now, with my jetlag in distinct control – its 3:30 AM for me – let me read it again and make a few comments here and there. (I just arrived in Haifa yesterday for a visit with Asher Peres; I'll be here a little over a week.)

• Why don't you write the following for your requested reference: C. A. Fuchs, “The Structure of Quantum Information,” available at […] [Of course, it is no longer available. Instead, see note to Carl Caves, dated 21 December 1997.]

• I like this view of physics as being like a jig-saw puzzle. Note a typo in this paragraph: you misspelled de Broglie.

Ruskai-ism 1:As far as I am aware, there is no evidence that either Heisenberg or Schrödinger was motivated by political or social factors.

Actually there is a pretty strong case that these two guys were indeed quite motivated by cultural factors to find an indeterministic element in physics. And that was long before quantum mechanics took a final form. Two important articles to read in this regard are:

1. P. Forman, “Weimar Culture, Causality, and Quantum Theory, 1918–1927: Adaptation by German Physicists and Mathematicians to a Hostile Intellectual Environment,” in Historical Studies in the Physical Sciences, Vol. 3, edited by R. McCormmach (U. Pennsylvania Press, Philadelphia, 1971), pp. 1–115.

2. […]

February 1999, “From Gleason to Pitowsky”

You'll note that I've forwarded you three emails from my correspondence with Itamar. [See note to Pitowsky dated 13 October 1998 (Other Letters).] I'd be keen to hear any further thoughts you might have on Gleason or Kochen and Specker. My email is always an open door when it comes to better understanding the quantum …

May 1999, “SD and KS” and “Even More Relevant”

How about we plan on my dropping by sometime during the week of May 24–28. You guys choose the exact dates (either one or two nights, as you said). I'll probably bring my wife and little girl down with me. They'll drop me off to play with you, and go off to play a little on their own.

I have a question about your rational-point anti-KS result. I'm starting to wonder how robust the idea is. The way you pose the problem is to consider the points on the sphere with rational cartesian coordinates. But are you sure that nothing changes under rather natural coordinate changes? For instance, imagine working in spherical coordinates for vectors on the unit sphere in R3. If we restrict ourselves to rational angles θ and φ, the cartesian coordinates so generated will almost always be irrational. And this discrete set of vectors is not just a rigid rotation of the ones you consider.

June 1997, “Painting My Stripes”

I am just returning from the library, where – completely by accident – I ran into and read an entertaining little article: A Peres, “Einstein, Gödel, Bohr,” Foundations of Physics15(2), 1985, 201–205. I was especially entertained by your closing line, “Any attempt to inject realism in physical theory is bound to lead to inconsistencies.” Do you have any other writings (beside the AJP paper with Zurek) where you expand upon the ideas presented in this?

In connection to this, I have another question. If I recall correctly, you once told me that you finally came to grips (“interpretationally,” that is) with quantum mechanics in 1986. (Perhaps I have the year wrong, but that's part of the question.) Would you be willing to expand upon that point? In particular, was your arrived-at comfort founded upon the line I quote above?

I suppose I ask these questions for several reasons, perhaps the strongest of which is a renewed dilettante interest in the foundations of quantum theory. Among other things, I am trying to plan (or dream up!) a talk for the Hull conference in September to be titled something like: “Quantum Information Theory's Implications for the Foundations of the Quantum as a Whole.”

My own pet “idea for an idea” (as Wheeler would say) is that the quantum theoretical description – along with its Hilbert-space structure – is forced upon us when the information we have about a physical situation/system is of a certain variety.

February 2000, “Your Project”

(Please remind me of our conversation in Italy. I remember your name, but I can't quite picture you.)

Your proposal sounds interesting: it wasn't too bold. Why don't we carry on just as you suggested. I'd be happy to supervise a project of yours (or perhaps collaborate)…as long as the work concerns something that interests me. Presently most of my interest is in quantifying different aspects of the trade-off between information and disturbance in quantum mechanics. This includes deriving various properties for some information functions that have little coverage in the existing literature so far. If we could find something that pleases us both in that general area, I think it'd be great. (Presently I'm not too interested in tackling entanglement questions: I can't see the forest for the trees in that subject.)

About collaborations, I only have one hard and fast rule at the outset: that is that author ordering be alphabetical. I place my form letter concerning this issue below. If you feel that this would hurt you, we can agree to keep MY role in the project in a strictly supervisory capacity.

Let me know what you think: I'm always within email contact.

17 January 2001, “First Contact”

I am writing this letter to find out if you are interested in attending a conference in Växjö, Sweden this summer titled “Quantum Theory: Reconsideration of Foundations.” I do this because I have read a few of your papers and parts of your book on Bohr, and I have been extremely impressed. By trade, I am a practitioner of the new field of quantum information theory and computing – and indeed the main session I am organizing at the Växjö meeting will be about quantum foundations reconsidered in that light – but your description of Bohr's thought meshes so well with what I'm seeking in physics that I'd like to get you there too. PLEASE NOTE that this is not exactly a real invitation yet: I am still lobbying for a more philosophical contingent at the meeting. But, I would very much like to hear your reaction just as soon as possible. If things work out, and you would like to come, I will recontact you very soon with further details.

The main reason I want you at our meeting is that I have this “madly optimistic” (Mermin called it) feeling that Bohrian–Paulian ideas will lead us to the next stage of physics. That is, that thinking about quantum foundations from their point of view will be the beginning of a new path, not the end of an old one.

02 June 1999, “Please Do”

You know what I'm really trying to get at here: I want the structure of quantum mechanics to be extremal for some kind of more abstract (i.e., QM-independent) information-disturbance problem. But the first thing that needs to be done is to pin down what quantum mechanics itself has to say.

Another thing that keeps coming to mind is that the assumptions behind Gleason's theorem might have something to do with the allegory I presented you with at Charlie and Theo's house. God said, “Ok I will make information gathering invasive, but not so invasive that you will be forever science-less.” The connection is this. I think one can split the assumption behind Gleason up into at least two parts: (1) The questions that can be asked of a quantum system only correspond to orthonormal bases over some vector space, with there being no good notion of measuring two distinct questions simultaneously (there being no good notion of an AND operation). And (2) It is the task of physical theory to give probabilities for the outcomes of those questions, but we can say at least this much about the probabilities. They are context-independent in the sense that, for a given outcome, it does not matter which physical question (which basis) we've associated it with. I.e., the probabilities can be assumed to be of the form of a frame function.

June 1996, “Cyclic Never-Endings”

Yes, yes, I like Montréal! Does that finally answer your question? The only thing I really miss about the states is having a swarm of good physicists around me.

El Jeffy 1:You know, I really miss listen to you babble about quantum mechanics. Nobody around any more to tell me that the moon isn't there when I'm not looking; you even had me believing for a while.

Damn, if I had only stayed there a bit longer…perhaps you would have remained a true believer!

August 1996, “Laws of Thought”

[Referring to the phrase “Quantum Mechanics is a Law of Thought.”:]

El Jeffy 2:I was just reading about the mathematician George Boole, the guy who invented Boolean Algebra. In 1854 he published a book called An Investigation of the Laws of Thought, on which are founded the mathematical theories of logic and probabilities. Is that where your lovely little phrase comes from?

Yes indeedy, that's exactly where it comes from. Quantum Mechanics is a Law of Thought. God I like the ring of that.

August 1996, “Plagiarists”

El Jeffy 3:I am disappointed in you. I never realized that you plagiarized that from poor ol' Boole. Can't think up your own catch phrase, eh?

I think the phrase bothers me far less when applied to something like Boolean algebra. It still gives me the heebies when you refer to real live physical systems with such terminology.

November 1997, “The Radical Eye”

Let me send you some prep work…while I'm procrastinating from things I really need to be doing. I'll convey the information in my favorite way: quoting/forwarding old emails!

The first, an old – kind of poorly written – note “Dreams of a More Ethereal Quantum,” concerns one aspect of the below-the-water stuff: the idea about distilling the MaxEntlooking part out of quantum theory. It doesn't say much more than I said in the proposal…but you can't expect too much, it's still just a vague program for a program. That stuff is at the bottom of the note; the early part of the note says a few – even more sketchy – things about Mermin and Quantum Probabilities…so I thought I would include it for that reason.

The only other readable thing (I think) I've written on this aspect of the below-the-water stuff is on pages 22–26 of my paper with Caves, “Quantum Information: How Much Information in a State Vector” (it's on quant-ph). These same pages also contain an extended discussion on probability.

The second note, “Rainy Mondays and Consistent Histories,” is a note I wrote Bob Griffiths concerning what I think of his Consistent History stuff. I think it says quite plainly and clearly what I think about probability in quantum mechanics. It's pretty self-contained (i.e., not referring to old conversations, etc.), so you might find it useful reading.

22 August 1999, “Your Article”

I am writing to let you know that I enjoyed your article quant-ph/9812050, “Why Quantum Mechanics is Hard to Understand,” very much. I was in particular very pleased with your Section IV “Dynamics vs. History.” It may interest you to know that Markus Fierz expressed a similar point of view in his article: “Does a physical theory comprehend an ‘objective, real, single process’?” in Observation and Interpretation in the Philosophy of Physics, edited by S. Körner (Dover, NY, 1957), pp. 93–96. I will append the full text below for your enjoyment. [See note to Rüdiger Schack, dated 29 August 1999.]

Most importantly, I would like to ask you for the final coordinates of your paper. Was it published somewhere? I would like to cite it in an upcoming paper.

28 August 1999, “Not Much”

I don't have a good idea about where to send your paper. I think chances are that AJP will be tougher than FP. In any case, it had an impact on me. My old advisor used to say of our joint papers, “If three people read this paper, we'll be doing OK!” If you know of two others with yours, you're probably doing OK.

I like this focus on the concept of “object.” I look forward to anything you might have to say about it. I have some slides that I use as a rallying cry to the idea that quantum information theory may have some impact on quantum foundations. The first is a list of axioms for quantum mechanics.

08 April 1996, “Hartle/Coleman Comments”

I like the grass fire analogy: I agree we should stomp it out if we can. However, as with most debates, I imagine we'll just have to wait until it falls by the wayside … or wait until someone gives it a name like the Modern Interpretation of Quantum Mechanics with Sugar on Top, so that everyone else will just discount it and move on.

Do you know whether Coleman has published any of this? Wojciech first told me about it at least a year ago (and told me I should contact Coleman concerning it to “help your career along”). If you have a copy of the paper, please send it my way. I've been at a little disadvantage in thinking about this because I cannot find my copy of Hartle's paper and the library has been shut down since Wednesday evening; I'll be able to get a new copy Tuesday morning. I know I liked the paper very much after turning semi-Bayesian because of what it said in words (even though I never agreed with the mathematics); however, it certainly did have a frequentist streak running through it … and that's what's bringing us back now.

I think what I'll do in this note is just write down various thoughts as they come to me; I've got a few pieces of junk spread out on my desk to prod me along.

September 1996, “Theorem II”

I encountered your “Ithaca Interpretation” paper this morning on the quant-ph archive … and, I must say, I've been walking around with a nice feeling since. There are some things in it that I like very much!

Your Theorem II (or a very slight variation of it) has indeed been proven before, by Bill Wootters. I'll attach the citations below along with part of a note Bill once sent me. (Interestingly, one can also make a cut between real, complex, and quaternionic Hilbert spaces based on such a theorem.)

About your Theorem I, if I am not mistaken, it also has a history that predates the references you gave. I believe Richard Jozsa once told me that a fellow named Hadjisavvas found it well before HJW and Gisin. If you are interested you can contact Richard directly; his e-mail address is.…

As I say, I enjoyed your paper very much! (Presently) I think my only point of departure in what you say is about the issue of “objective probability.” Carl Caves and I (in still another paper cited below which can be found on quant-ph, pp. 22–26 in particular) have advocated the view that even the probabilities of quantum theory should be interpreted in a Bayesian or subjective way. We prefer to say that it is the “indeterminism” of quantum mechanics that is “objective,” and not the probabilities themselves.