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\begin{document}

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\section*{Bub-Clifton theorem}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebraic_quantum_field_theory}{}\paragraph*{{Algebraic Quantum Field Theory}}\label{algebraic_quantum_field_theory}

[[!include AQFT and operator algebra contents]]

\hypertarget{philosophy}{}\paragraph*{{Philosophy}}\label{philosophy}

[[!include philosophy - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The two fundamental [[no-go theorems]] for [[hidden variable theory|hidden variable]] reconstructions of the [[quantum mechanics|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 \emph{Bub-Clifton-Halvorson theorem} (\hyperlink{BubClifton96}{Bub-Clifton 96}, \hyperlink{CliftonBubHalvorson03}{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 \hyperlink{Bub09}{Bub 09, pages 1-2}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[interpretation of quantum mechanics]]


\item [[beable]]



\end{itemize}
Other theorems about the foundations and [[interpretation of quantum mechanics]] include:

\begin{itemize}%
\item [[order-theoretic structure in quantum mechanics]]

\begin{itemize}%
\item [[Kochen-Specker theorem]]


\item [[Alfsen-Shultz theorem]]


\item [[Harding-Döring-Hamhalter theorem]]



\end{itemize}

\item [[Fell's theorem]]


\item [[Gleason's theorem]]


\item [[Wigner theorem]]


\item [[Bell's theorem]]


\item [[Kadison-Singer problem]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Jeffrey Bub]], Rob Clifton, \emph{A Uniqueness Theorem for ``No Collapse'' Interpretations of Quantum Mechanics}, Studies in the History and Philosophy of Modern Physics 27, 181-219. (1996)


\item [[Jeffrey Bub]], Rob Clifton, [[Sheldon Goldstein]], \emph{Revised Proof of the Uniqueness Theorem for `No Collapse' Interpretations of Quantum Mechanics}, Studies in History and Philosophy of Modern Physics 31 (2000) 95 (\href{http://arxiv.org/abs/quant-ph/9910097}{arXiv:quant-ph/9910097})


\item Rob Clifton, [[Jeffrey Bub]], [[Hans Halvorson]], \emph{Characterizing Quantum Theory in Terms of Information-Theoretic Constraints}, Foundations of Physics 33 (2003), p 1561 (\href{https://arxiv.org/abs/quant-ph/0211089}{arXiv:quant-ph/0211089})


\item [[Jeffrey Bub]], \emph{Bub-Clifton Theorem}, in Greenberger D., Hentschel K., Weinert F. (eds.) \emph{Compendium of Quantum Physics} Springer 2009, pp 84-86 (\href{https://link.springer.com/chapter/10.1007%2F978-3-540-70626-7_25}{web})



\end{itemize}
A broader textbook discussion is in

\begin{itemize}%
\item [[Jeffrey Bub]], \emph{Interpreting the quantum world}, Cambridge University Press, Aug 26, 1999

\end{itemize}
An interpretation in [[topos theory]] is proposed in

\begin{itemize}%
\item Kunji Nakayama, \emph{Topos-Theoretic Extension of a Modal Interpretation of Quantum Mechanics}, Int. J. Theor. Phys. 47:2065-2094, 2008 (\href{https://arxiv.org/abs/0711.2200}{arXiv:0711.2200})

\end{itemize}
See also

\begin{itemize}%
\item Jonathan Bain, \emph{Physics from Quantum Information} (\href{http://faculty.poly.edu/~jbain/physinfocomp/lectures/10.PhysicsfromQI.pdf}{pdf}), chapter 10 in \emph{\href{http://faculty.poly.edu/~jbain/physinfocomp/}{Physics, Information and Computation}}


\item Wikipedia, \emph{\href{https://de.wikipedia.org/wiki/CBH-Theorem}{CBH-Theorem}}



\end{itemize}
[[!redirects Bub-Clifton-Halvorson theorem]]

[[!redirects CBH-theorem]]



\end{document}