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Letters to Joseph Renes

Christopher A. Fuchs

Published Jan 6, 2011• 0 citations

Abstract

June 2000, “References for Gleason” Some relevant references. The important ones are Gleason's paper, Cooke et al .'s paper, and Pitowsky's paper. R. Cooke, M. Keane, and W. Moran, “An Elementary Proof of Gleason's Theorem,” Math. Proc. Camb. Phil. Soc. 98 , 117–128 (1981). A. M. Gleason, “Measures on the Closed Subspaces of a Hilbert Space,” J. Math. Mech. 6 , 885–894 (1957). I. Pitowsky, “George Boole's ‘Conditions of Possible Experience’ and the Quantum Puzzle,” Brit. J. Phil. Sci. 45 , 95–125 (1994). I. Pitowsky, “Infinite and Finite Gleason's Theorems and the Logic of Indeterminacy,” J. Math. Phys. 39 , 218–228 (1998). 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.

DOI: 10.1017/cbo9780511762789.029

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June 2000, “References for Gleason” Some relevant references. The important ones are Gleason's paper, Cooke et al .'s paper, and Pitowsky's paper. R. Cooke, M. Keane, and W. Moran, “An Elementary Proof of Gleason's Theorem,” Math. Proc. Camb. Phil. Soc. 98 , 117–128 (1981). A. M. Gleason, “Measures on the Closed Subspaces of a Hilbert Space,” J. Math. Mech. 6 , 885–894 (1957). I. Pitowsky, “George Boole's ‘Conditions of Possible Experience’ and the Quantum Puzzle,” Brit. J. Phil. Sci. 45 , 95–125 (1994). I. Pitowsky, “Infinite and Finite Gleason's Theorems and the Logic of Indeterminacy,” J. Math. Phys. 39 , 218–228 (1998). 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.

We formalize in Lean 4/Mathlib a finite-dimensional route to Gleason’s theorem based on Cooke–Keane–Moran and Richman–Bridges, with Dvureˇcenskij’s reduction from complex spaces to real three-dimensional sections. Cooke–Keane–Moran obtain the extremal step by compactness of a product space; in Lean we realize their accumulation point by an ultrafilter. More substantially, the opening step of their general-case argument in §7 invokes ingredients that do not, as stated, supply the required minimizer orthogonal to a maximizer. We give the missing two-frame argument explicitly and machine-check the resulting theorem for Hn = Cn,n ≥ 3. On top of this representation layer we formalize the one-directional Uhlhorn–ˇSemrl theorem as an application of Gleason and Wigner. The development also includes Busch’s effect-measure theorem, a projective weight theorem whose local binary-split formulation generates finite refinement coherence, and a finite Naimark interface with a strict classification of binary implementations. Classical failures at dimension two are formalized at the exact input types of the corresponding theorems, while the effect-domain route recovers the qubit case. Together these results exhibit a distinct proof route to representation, one local justification exposed by formalization, and typed boundaries separating projection, effect, symmetry and refinement interfaces.

Abstract Gleason’s theorem (Gleason 1957 J. Math. Mech. 6 885) is an important result in the foundations of quantum mechanics, where it justifies the Born rule as a mathematical consequence of the quantum formalism. Formally, it presents a key insight into the projective geometry of Hilbert spaces, showing that finitely additive measures on the projection lattice extend to positive linear functionals on the algebra of bounded operators . Over many years, and by the effort of various authors, the theorem has been broadened in its scope from type I to arbitrary von Neumann algebras (without type I 2 factors). Here, we prove a generalisation of Gleason’s theorem to composite systems. To this end, we strengthen the original result in two ways: first, we extend its scope to dilations in the sense of Naimark (1943 Dokl. Akad. Sci. SSSR 41 359) and Stinespring (1955 Proc. Am. Math. Soc. 6 211) and second, we require consistency with respect to dynamical correspondences on the respective (local) algebras in the composition (Alfsen and Shultz 1998 Commun. Math. Phys. 194 87). We show that neither of these conditions changes the result in the single system case, yet both are necessary to obtain a generalisation to bipartite systems.

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.