Contents

Idea

The two fundamental no-go theorems for hidden variable reconstructions of the quantum statistics, the Kochen-Specker theorem and Bell's theorem, may be formulated as results about the impossibility of associating a classical probability space with a quantum system, when certain constraints are placed on the probability measure.

The Bub-Clifton-Halvorson theorem (Bub-Clifton 96, Clifton-Bub-Halvorson 03) on the other hand is a positive result about the possibility of associating a classical probability space with a quantum system in a given state.

Given a Hilbert space $H$ and a quantum observable represented by a Hermitean operator on this space, and given a pure state, the Bub-Clifton theorem characterizes a maximal sub-lattice of the Birkhoff-vonNeumann Hilbert lattice of subspace of $H$ (the “quantum logic” of $H$) such that there are sufficiently many yes-no questions on the elements in the lattice to recover all the probabilities induced by the given pure state to compatible sets of projections (classical contexts).

A useful summary is in Bub 09, pages 1-2.

Related concepts

• interpretation of quantum mechanics

• beable

Other theorems about the foundations and interpretation of quantum mechanics include:

• order-theoretic structure in quantum mechanics

  • Kochen-Specker theorem

  • Alfsen-Shultz theorem

  • Harding-Döring-Hamhalter theorem

• Fell's theorem

• Gleason's theorem

• Wigner theorem

• Bell's theorem

• Kadison-Singer problem

References

• Jeffrey Bub, Rob Clifton, A Uniqueness Theorem for “No Collapse” Interpretations of Quantum Mechanics, Studies in the History

  and Philosophy of Modern Physics 27, 181-219. (1996)

• Jeffrey Bub, Rob Clifton, Sheldon Goldstein, Revised Proof of the Uniqueness Theorem for ‘No Collapse’ Interpretations of Quantum Mechanics, Studies in History and Philosophy of Modern Physics 31 (2000) 95 (arXiv:quant-ph/9910097)

• Rob Clifton, Jeffrey Bub, Hans Halvorson, Characterizing Quantum Theory in Terms of Information-Theoretic Constraints, Foundations of Physics 33 (2003), p 1561 (arXiv:quant-ph/0211089)

• Jeffrey Bub, Bub-Clifton Theorem, in Greenberger D., Hentschel K., Weinert F. (eds.) Compendium of Quantum Physics

  Springer 2009, pp 84-86 (web)

A broader textbook discussion is in

• Jeffrey Bub, Interpreting the quantum world, Cambridge University Press, Aug 26, 1999

An interpretation in topos theory is proposed in

• Kunji Nakayama, Topos-Theoretic Extension of a Modal Interpretation of Quantum Mechanics, Int. J. Theor. Phys. 47:2065-2094, 2008 (arXiv:0711.2200)

See also

• Jonathan Bain, Physics from Quantum Information (pdf), chapter 10 in Physics, Information and Computation

• Wikipedia, CBH-Theorem