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\section*{quantum probability theory}

[[!redirects quantum probability]] [[!redirects quantum probability]]

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

\hypertarget{measure_and_probability_theory}{}\paragraph*{{Measure and probability theory}}\label{measure_and_probability_theory}

[[!include measure theory - contents]]

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

[[!include AQFT and operator algebra contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{AsClassicalProbabilityInBohrTopos}{As classical probability theory internal to a Bohr topos}\dotfill \pageref*{AsClassicalProbabilityInBohrTopos} \linebreak
\noindent\hyperlink{ConditionalExpectationAndWaveFunctionCollapse}{Conditional expectation and Wave function collapse}\dotfill \pageref*{ConditionalExpectationAndWaveFunctionCollapse} \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}

In [[probability theory]], the concept of \emph{noncommutative probability space} or \emph{quantum probability space} is the generalization of that of \emph{[[probability space]]} as the concept of ``space'' is generalized to [[non-commutative geometry]].

The basic idea is to encode a would-be [[probability space]] [[Isbell duality|dually]] in its [[algebra of functions]] $\mathcal{A}$, typically regarded as a [[star algebra]], and encode the [[probability measure]] as a [[state on a star-algebra|state on this star algebra]]

\begin{displaymath}
\langle
    -
  \rangle
  \;\colon\;
  \mathcal{A}
    \longrightarrow
  \mathbb{C}
  \,.
\end{displaymath}
Hence this primarily [[axiom|axiomatizes]] the concept of \emph{[[expectation values]]} $\langle A\rangle$ (\hyperlink{Segal65}{Segal 65}, \hyperlink{Whittle92}{Whittle 92}) while leaving the nature of the underlying [[probability measure]] implicit (in contrast to the classical formalization of [[probability theory]] by [[Andrey Kolmogorov]]).

Often $\mathcal{A}$ is assumed/required to be a [[von Neumann algebra]] (e.g. \hyperlink{Kuperberg05}{Kuperberg 05, section 1.8}). Often $\mathcal{A}$ is taken to be the full algebra of [[bounded operators]] on some [[Hilbert space]] (e.g. \hyperlink{Attal}{Attal, def. 7.1}).

In [[quantum physics]], $\mathcal{A}$ is an [[algebra of observables]] (or a [[local net of observables|local net]] thereof) and $\langle (-)\rangle$ is a particular [[quantum state]], for instance a [[vacuum state]].

The formulation of [[non-perturbative quantum field theory]] from the algebraic perspective of quantum probability is known as \emph{[[algebraic quantum field theory]]} ([[AQFT]]).

The formulation of [[perturbative quantum field theory]] from the algebraic perspective of quantum probability is known as \emph{[[perturbative algebraic quantum field theory]]} ([[pAQFT]]).

The sentiment that [[quantum physics]] \emph{is} quantum probability theory is also referred to as the \emph{[[Bayesian interpretation of quantum mechanics]]} (``QBism'').

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{AsClassicalProbabilityInBohrTopos}{}\subsubsection*{{As classical probability theory internal to a Bohr topos}}\label{AsClassicalProbabilityInBohrTopos}

The idea that

\begin{quote}%
[[quantum probability]] is ``just as'' classical [[probability theory]] but generalized to [[non-commutative geometry|non-commutative]] [[probability spaces]], hence, for [[quantum physics]], to quantized [[phase spaces]]


\end{quote}
may be made precise and fully manifest by understanding quantum probability theory as being classical [[probability theory]] \emph{[[internalization|internal]]} to the \emph{[[Bohr topos]]} of the given [[quantum mechanical system]].

For details see at \emph{[[Bohr topos]]} the section \emph{\href{Bohr+topos#KinematicsOnBohrTopos}{Kinematics in a Bohr topos}}.

For going deeper, see at \emph{[[order-theoretic structure in quantum mechanics]]}.

\hypertarget{ConditionalExpectationAndWaveFunctionCollapse}{}\subsubsection*{{Conditional expectation and Wave function collapse}}\label{ConditionalExpectationAndWaveFunctionCollapse}

Quantum probability theory shows that ``[[wave function collapse]]'' is just part of the formula for [[conditional expectation values]] in [[quantum probability theory]] (e.g. \hyperlink{Kuperberg05}{Kuperberg 05, section 1.2}, \hyperlink{Yuan12}{Yuan 12}):

Let $(\mathcal{A},\langle -\rangle)$ be a [[quantum probability space]], hence a [[complex numbers|complex]] [[star algebra]] $\mathcal{A}$ of [[quantum observables]], and a [[state on a star-algebra]] $\langle -\rangle \;\colon\; \mathcal{A} \to \mathbb{C}$.

This means that for $A \in \mathcal{A}$ any [[observable]], its \emph{[[expectation value]]} in the given [[state on a star-algebra|state]] is

\begin{displaymath}
\mathbb{E}(A) \;\coloneqq\;  \langle A \rangle \in \mathbb{C}
  \,.
\end{displaymath}
More generally, if $P \in \mathcal{A}$ is a [[real part|real]] [[idempotent]]/[[projector]]

\begin{equation}
P^\ast = P
  \,,
  \phantom{AAA}
  P P = P
\label{RealIdempotent}\end{equation}
thought of as an event, then for any observable $A \in \mathcal{A}$ the [[conditional expectation value]] of $A$, conditioned on the observation of $P$, is (e.g. \hyperlink{RedeiSummers06}{Redei-Summers 06, section 7.3}, see also \hyperlink{FroehlichSchubnel15}{Fröhlich-Schubnel 15, (5.49)}, \href{Froehlich19}{Fröhlich 19 (45)})

\begin{equation}
\mathbb{E}(A \vert P)
  \;\coloneqq\;
  \frac{
    \left \langle P A P \right\rangle
  }{
    \left\langle P \right\rangle
  }
  \,.
\label{ConditionalExpectation}\end{equation}
Now assume a [[star-representation]] $\rho \;\colon\; \mathcal{A} \to End(\mathcal{H})$ of the [[algebra of observables]] by [[linear operators]] on a [[Hilbert space]] $\mathcal{H}$ is given, and that the state $\langle -\rangle$ is a [[pure state]], hence given by an [[vector]] $\psi \in \mathcal{H}$ (``[[wave function]]'') via the [[Hilbert space]] [[inner product]] $\langle (-), (-)\rangle \;\colon\; \mathcal{H} \otimes \mathcal{H} \to \mathbb{C}$ as

\begin{displaymath}
\begin{aligned}
    \langle A \rangle
    & \coloneqq
    \left\langle\psi
      \vert A \vert \psi
    \right\rangle
    \\
    & \coloneqq
    \left\langle\psi,
      A \psi
    \right\rangle
  \end{aligned}
  \,.
\end{displaymath}
In this case the expression for the [[conditional expectation value]] \eqref{ConditionalExpectation} of an observable $A$ conditioned on an idempotent observable $P$ becomes (notationally suppressing the [[representation]] $\rho$)

\begin{displaymath}
\begin{aligned}
    \mathbb{E}(A\vert P)
    & =
    \frac{
      \left\langle
        \psi \vert P A P\vert \psi
      \right\rangle
    }{
      \left\langle 
        \psi \vert P \vert \psi
      \right\rangle
    }
    \\
    & = 
    \frac{
      \left\langle 
        P \psi \vert A \vert P \psi
      \right\rangle
    }{
     \left\langle 
        P \psi \vert P \psi 
     \right\rangle
    }  
    \,,
  \end{aligned}
\end{displaymath}
where in the last step we used \eqref{ConditionalExpectation}.

This says that \emph{assuming} that $P$ has been observed in the [[pure state]] $\vert \psi\rangle$, then the corresponding [[conditional expectation values]] are the same as actual [[expectation values]] but for the new pure state $\vert P \psi \rangle$.

This is the statement of ``[[wave function collapse]]'':

\begin{displaymath}
\vert \psi \rangle \mapsto P \vert \psi \rangle
  \,.
\end{displaymath}
The original [[wave function]] is $\psi \in \mathcal{H}$, and after observing $P$ it ``collapses'' to $P \psi \in \mathcal{H}$ (up to normalization).

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

\begin{itemize}%
\item [[probability amplitude]]


\item [[quantum statistical mechanics]]


\item [[Bayesian interpretation of quantum mechanics]]


\item [[quantum logic]]


\item [[Bohr topos]]



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

The axiomatization of [[probability theory]] in terms of the concept of [[expectation values]] (instead of [[probability measures]]) is amplified in

\begin{itemize}%
\item [[Irving Segal]], \emph{Algebraic integration theory}, Bull. Amer. Math. Soc. Volume 71, Number 3, Part 1 (1965), 419-489 (\href{https://projecteuclid.org/euclid.bams/1183526903}{Euclid})


\item [[Peter Whittle]], \emph{Probability via expectation}, Springer 1992



\end{itemize}
Gentle exposition of the basics:

\begin{itemize}%
\item Jonathan Gleason, \emph{The $C^*$-algebraic formalism of quantum mechanics}, 2009 ([[Gleason09.pdf:file]], [[GleasonAlgebraic.pdf:file]])

\end{itemize}
Further introduction to quantum probability theory:

\begin{itemize}%
\item [[Greg Kuperberg]], \emph{A concise introduction to quantum probability, quantum mechanics, and quantum computation}, 2005 (\href{http://www.math.ucdavis.edu/~greg/intro-2005.pdf}{pdf})


\item [[Miklos Redei]], [[Stephen Summers]], \emph{Quantum Probability Theory} (\href{https://arxiv.org/abs/quant-ph/0601158}{arXiv:quant-ph/0601158})


\item S. Attal, \emph{Quantum probability} (\href{http://math.univ-lyon1.fr/~attal/Quantum_Probability.pdf}{pdf})


\item [[Qiaochu Yuan]], \emph{\href{https://qchu.wordpress.com/2012/08/18/noncommutative-probability/}{Noncommutative probability}}, 2012


\item [[Qiaochu Yuan]], \emph{\href{https://qchu.wordpress.com/2012/09/09/finite-noncommutative-probability-the-born-rule-and-wave-function-collapse/}{Finite noncommutative probability, the Born rule, and wave function collapse}}, 2012



\end{itemize}
Monographs:

\begin{itemize}%
\item [[Jürg Fröhlich]], B. Schubnel, \emph{Quantum Probability Theory and the Foundations of Quantum Mechanics}. In: Blanchard P., Fröhlich J. (eds.) \emph{The Message of Quantum Science}. Lecture Notes in Physics, vol 899. Springer 2015 (\href{https://arxiv.org/abs/1310.1484}{arXiv:1310.1484}, \href{https://doi.org/10.1007/978-3-662-46422-9_7}{doi:10.1007/978-3-662-46422-9\_7})


\item [[Jürg Fröhlich]], \emph{The structure of quantum theory}, Chapter 6 in \emph{The quest for laws and structure}, EMS 2016 (\href{https://www.researchgate.net/publication/308595814_The_Quest_for_Laws_and_Structure}{doi}, \href{https://www.ems-ph.org/books/show_abstract.php?proj_nr=207&vol=1&rank=8}{doi:10.4171/164-1/8})


\item [[Klaas Landsman]], \emph{Foundations of quantum theory -- From classical concepts to Operator algebras}, Springer Open 2017 (\href{https://link.springer.com/content/pdf/10.1007%2F978-3-319-51777-3.pdf}{pdf})



\end{itemize}
[[!redirects quantum probability theories]]

[[!redirects quantum probability]] [[!redirects quantum probabilities]]

[[!redirects quantum probability space]] [[!redirects quantum probability spaces]]

[[!redirects noncommutative probability]] [[!redirects noncommutative probabilities]] [[!redirects non-commutative probability]] [[!redirects non-commutative probabilities]]

[[!redirects noncommutative probability space]] [[!redirects noncommutative probability spaces]] [[!redirects non-commutative probability space]] [[!redirects non-commutative probability spaces]]

[[!redirects noncommutative probability theory]] [[!redirects noncommutative probability theories]] [[!redirects non-commutative probability theory]] [[!redirects non-commutative probability theories]]



\end{document}