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\section*{Gleason&#39;s theorem}

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

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{aqft}{}\paragraph*{{AQFT}}\label{aqft}

[[!include AQFT and operator algebra contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{implications_for_quantum_logic}{Implications for Quantum Logic}\dotfill \pageref*{implications_for_quantum_logic} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{the_theorem}{The Theorem}\dotfill \pageref*{the_theorem} \linebreak
\noindent\hyperlink{classical_gleasons_theorem}{Classical Gleason's Theorem}\dotfill \pageref*{classical_gleasons_theorem} \linebreak
\noindent\hyperlink{extension_to_certain_types_of_von_neumann_algebras}{Extension to Certain Types of von Neumann Algebras}\dotfill \pageref*{extension_to_certain_types_of_von_neumann_algebras} \linebreak
\noindent\hyperlink{gleasons_theorem_for_povms}{Gleason's Theorem for POVMs}\dotfill \pageref*{gleasons_theorem_for_povms} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{counterexample_for_dimension_two}{Counterexample For Dimension Two}\dotfill \pageref*{counterexample_for_dimension_two} \linebreak
\noindent\hyperlink{related_theorems}{Related theorems}\dotfill \pageref*{related_theorems} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

The classical \emph{Gleason theorem} says that a [[state on a star-algebra|state]] on the [[C\emph{-algebra]] $\mathcal{B}(\mathcal{H})$ of all [[bounded operators]] on a [[Hilbert space]] is uniquely described by the values it takes on the orthogonal [[projections]] $\mathcal{P}$, if the dimension of the [[Hilbert space]] $\mathcal{H}$ is not 2.}

In other words: every [[quasi-state]] is already a state if $dim(H) \gt 2$.

It is possible to extend the theorem to certain types of [[von Neumann algebras]] (e.g. obviously [[von Neumann algebra factor|factors]] of type $I_2$ have to be excluded).

Gleason's theorem is also valid for real and quaternionic Hilbert spaces as proved by Varadarajan in 1968. A gap of that proof has been fixed in 2018 by V.Moretti and M.Oppio.

\hypertarget{implications_for_quantum_logic}{}\subsubsection*{{Implications for Quantum Logic}}\label{implications_for_quantum_logic}

Roughly, Gleason's theorem says that ``a quantum state is completely determined by only knowing the answers to all of the possible yes/no questions''.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\begin{defn}
\label{}\hypertarget{}{}
Let $\rho: \mathcal{P} \to [0, 1]$ such that for every finite family $\{ P_1, ..., P_n: P_i \in \mathcal{P} \}$ of pairwise orthogonal projections we have $\rho(\sum_{i=1}^n P_i) = \sum_{i=1}^n \rho(P_i)$, then $\rho$ is a \textbf{finitely additive measure} on $\mathcal{P}$.

If the family is not finite, but countable, then $\rho$ is a \textbf{sigma-finite measure}.

\end{defn}
\hypertarget{the_theorem}{}\subsection*{{The Theorem}}\label{the_theorem}

\hypertarget{classical_gleasons_theorem}{}\subsubsection*{{Classical Gleason's Theorem}}\label{classical_gleasons_theorem}

\begin{theorem}
\label{}\hypertarget{}{}
If $dim(\mathcal{H}) \neq 2$ then each finitely additive measure on $\mathcal{P}$ can be uniquely extended to a state on $\mathcal{B}(\mathcal{H})$. Conversly the restriction of every state to $\mathcal{P}$ is a finitley additive measure on $\mathcal{P}$.

The same holds for sigma-finite measures and [[normal state|normal states]]: Every sigma-finite measure can be extended to a normal state and every normal state restricts to a sigma-finite measure.

\end{theorem}
\hypertarget{extension_to_certain_types_of_von_neumann_algebras}{}\subsubsection*{{Extension to Certain Types of von Neumann Algebras}}\label{extension_to_certain_types_of_von_neumann_algebras}

\ldots{}

\hypertarget{gleasons_theorem_for_povms}{}\subsubsection*{{Gleason's Theorem for POVMs}}\label{gleasons_theorem_for_povms}

In [[quantum information theory]], one often considers positive operator-valued measures ([[POVM]]s) instead of [[Hermitian operators]] as [[observables]]. While a Hermitian operator is given by a family of [[projection operator]]s $P_i$ such that $\sum_i P_i = 1$, a POVM is given more generally by any family of positive-semidefinite operators $E_i$ such that $\sum_i E_i = 1$.

In the analog of Gleason's Theorem for POVMs, therefore, we start with $\rho\colon \mathcal{E} \to [0,1]$, where $\mathcal{E}$ is the space of all positive-semidefinite operators. Then if $\sum_i \rho(E_i) = 1$ whenever $\rho(\sum_i E_i) = 1$, the theorem states that $\rho$ has a unique extension to a mixed [[quantum state]].

As a theorem, Gleason's Theorem for POVMs is much weaker than the classical Gleason's Theorem, since we must begin with $\rho$ defined on a much larger space of operators. However, some content does remain, since we have not assumed any continuity properties of $\rho$. Also, Gleason's Theorem for POVMs has a much simpler proof, which works regardless of the dimension.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{counterexample_for_dimension_two}{}\subsubsection*{{Counterexample For Dimension Two}}\label{counterexample_for_dimension_two}

See example 8.1 in the book by Parthasarathy (see references). Our Hilbert space is $\mathbb{R}^2$. Projections $P$ on it are either identical zero, the identity, or projections on a one dimensional subspace, so that these $P$ can be written in the [[bra-ket notation]] as

\begin{displaymath}
P = {|u \rangle} {\langle u|}
\end{displaymath}
with a unit vector $u$, i.e. $u \in \mathbb{R}^2, {\|u\|} = 1$. In this finite dimensional case sigma-finite and finite are equivalent, and a finite probability measure is equivalent to a (complex valued) function such that

\begin{displaymath}
f(c u) = f(u)
\end{displaymath}
\begin{displaymath}
\sum_i f(u_i) = 1
\end{displaymath}
for every scalar $c$ of modulus one, every unit vector $u$ and every orthonormal basis $\{u_1, u_2\}$. If there is a state that extends such a measure and therefore restricts to such a measure on projections, there would be a linear operator $T$ such that

\begin{displaymath}
f(u) = {\langle u | T u \rangle}
\end{displaymath}
for all unit vectors $u$.

It turns out however that the conditions imposed on $f$ are not enough in two dimensions to enforce this kind of linearity of $f$. Heuristically, in three dimensions there are more rotations than in two, therefore the ``rotational invariance'' of (the conditions imposed on) $f$ is more restrictive in three dimensions than it is in two dimensions.

In two dimensions, choose a function $g$ on $[0, \frac{\pi}{2})$ such that $0 \le g(\theta) \le 1$ everywhere. There are no further restrictions, that is $g$ need not be continuous, for example. Now we can define a probability measure on the projections by

\begin{displaymath}
f(u) = 
\begin{cases}

  g(\theta) \; \; \text{for} \; \; 0 \le \theta \lt \frac{\pi}{2} \\
  1 - g(\theta -  \frac{\pi}{2}) \; \; \text{for} \; \; \frac{\pi}{2} \le \theta \lt \pi \\
  f(-u) \; \; \text{as defined in the first two items, else}
\end{cases}
\end{displaymath}
This probability measure will in general not extend to a state.

\hypertarget{related_theorems}{}\subsection*{{Related theorems}}\label{related_theorems}

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 [[Wigner theorem]]


\item [[Bell's theorem]]


\item [[Bub-Clifton theorem]]


\item [[Kadison-Singer problem]]



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

Gleason's original paper outlining the theorem is

\begin{itemize}%
\item [[Andrew Gleason]], \emph{Measures on the closed subspaces of a Hilbert space}, Journal of Mathematics and Mechanics, Indiana Univ. Math. J. 6 No. 4 (1957), 885--893 (\href{http://www.iumj.indiana.edu/IUMJ/FULLTEXT/1957/6/56050}{web})

\end{itemize}
A standard textbook exposition of the theorem and its meaning is

\begin{itemize}%
\item [[Huzihiro Araki]], \emph{[[Mathematical Theory of Quantum Fields]]} ,

\end{itemize}
where it appears as theorem 2.3 (without proof).

A monograph stating and proving both the classical theorem and extensions to [[von Neumann algebras]] is

\begin{itemize}%
\item Jan Hamhalter, \emph{Quantum measure theory} (\href{http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1038.81003&format=complete}{ZMATH entry})

\end{itemize}
The classical theorem is proved also in this monograph:

\begin{itemize}%
\item K. R. Parthasarathy, \emph{An Introduction to Quantum Stochastic Calculus} (\href{http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0751.60046&format=complete}{ZMATH})

\end{itemize}
Gleason's Theorem for POVMs is proved here:

\begin{itemize}%
\item Paul Busch, \emph{Quantum states and generalized observables: a simple proof of Gleason's theorem}; (1999) (\href{http://arxiv.org/abs/quant-ph/9909073}{arXiv})

\end{itemize}
The failure of Gleason's theorem for \emph{classical} states (on [[Poisson algebra]]s) is discussed in

\begin{itemize}%
\item Michael Entov, Leonid Polterovich, \emph{Symplectic quasi-states and semi-simplicity of quantum homology} (\href{http://arxiv.org/abs/0705.3735}{arXiv}).

\end{itemize}
Gleason's Theorem proved for real, complex and quaternionic Hilbert spaces using the notion of real trace.

\begin{itemize}%
\item [[Valter Moretti]], Marco Oppio, \emph{The correct formulation of Gleason's theorem in quaternionic Hilbert spaces}, Ann. Henri Poincaré 19 (2018), 3321-3355 (\href{http://arxiv.org/abs/1803.06882}{arXiv:1803.06882})

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia on \href{http://en.wikipedia.org/wiki/Gleason%27s_theorem}{Gleason's theorem}

\end{itemize}
[[!redirects Gleason's theorem]] [[!redirects Gleason's theorem]] [[!redirects Gleason theorem]] [[!redirects Gleason's Theorem]] [[!redirects Gleason's Theorem]] [[!redirects Gleason Theorem]]



\end{document}