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\begin{document}

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\section*{pure state}

\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{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

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

[[!include AQFT and operator algebra contents]]

\hypertarget{pure_and_mixed_states}{}\section*{{Pure and mixed states}}\label{pure_and_mixed_states}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{special_cases}{Special cases}\dotfill \pageref*{special_cases} \linebreak
\noindent\hyperlink{classical_versus_quantum}{Classical versus quantum}\dotfill \pageref*{classical_versus_quantum} \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}

A \emph{pure state} is a [[state on a star-algebra]] which is an extremal point in the [[convex set]] of all states.

In [[physics]], recall that a [[state]] of a [[physical system]] is (in the [[Bayesian interpretation of quantum mechanics|Bayesian interpretation]]) a specification of the information that one might have about the system (typically relative to some fixed background information). States form (at least) a [[poset]] where $\psi \leq \phi$ means that $\phi$ contains all of the information of $\psi$ (and possibly more). A \textbf{pure state} is a [[maximal element]] of this poset: a state that specifies as much information as possible about the system. A \textbf{mixed state} is a state that is not pure.

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

Fairly generally, a physical system has a complex [[C\emph{-algebra]] $\mathcal{A}$ of [[observables]] (or more generally a [[unital algebra|unital]] [[star algebra]]) and a \textbf{[[state on a star-algebra|state]]} is a positive-semidefinite [[linear function]] $\rho\colon \mathcal{A} \to \mathbb{C}$ such that $\rho(1) = 1$. We might write $\rho$ as a convex-[[linear combination]] of two other states:}

\begin{equation}
\rho = a \sigma + b \tau ,
\label{combo}\end{equation}
where necessarily $0 \leq a, b \leq 1$ and $a + b = 1$.

The state $\rho$ is \textbf{pure} if, whenever \eqref{combo} holds for $\sigma \neq \tau$, then either $a = 0$ (hence $b = 1$ and $\rho = \tau$) or $a = 1$ (hence $b = 0$ and $\rho = \sigma$); conversely, $\rho$ is \textbf{mixed} if we ever have \eqref{combo} for $\sigma \neq \tau$ and $0 \lt a \lt 1$ (hence $0 \lt b \lt 1$ and $\rho \neq \sigma, \tau$).

Really, this definition makes sense as long as the states form a [[convex space]].

To define when one state gives at least as much information as another (the [[partial order]] from the Idea section), let $\rho \leq \sigma$ mean that the [[mutual information]] $I(\rho,\sigma)$ equals the [[entropy]] $H(\rho)$, or equivalently that the [[conditional entropy]] $H(\rho|\sigma)$ is zero. (In the classical case, this partial order is attributed to \hyperlink{Shannon1953}{Shannon (1953)}, which I have not read, by \hyperlink{LiChong2011}{Li \& Chong (2011)}, which I have only skimmed.) The [[maximal elements]] under this partial order should be precisely the pure states, but the direct definition of pure states is much simpler.

I need to check whether these are equivalent on any $C^*$-algebra. ---Toby

\hypertarget{special_cases}{}\subsection*{{Special cases}}\label{special_cases}

If $A$ is the algebra of continuous complex-valued functions on some [[compactum]] $X$, then the \textbf{pure states} on $A$ correspond precisely to the [[points]] in $X$; so pure states here are the states of [[classical mechanics]] (at least for a compact [[phase space]]). The mixed states, however, correspond more generally to [[Radon measure|Radon]] [[probability measures]] on $X$, with the pure states as the [[Dirac delta measure]]s.

On the other hand, if $A$ is the algebra of all [[bounded operators]] on some [[Hilbert space]] $H$, then the pure states on $A$ correspond precisely to the rays in $H$, as is usual in [[quantum mechanics]]. The mixed states, however, correspond more generally to [[density matrices]] on $H$, with the pure states those matrices of the form ${|\psi\rangle}{\langle\psi|}$ for some unit vector ${|\psi\rangle}$.

\hypertarget{classical_versus_quantum}{}\subsection*{{Classical versus quantum}}\label{classical_versus_quantum}

In any case, a pure state is a state of maximal information, while a mixed state is a state with less than maximal information. In the classical case, we may say that a pure state is a state of \emph{complete} information, but this does not work in the quantum case; from the perspective of the information-theoretic or Bayesian interpretation of quantum physics, this inability to have complete information, even when having maximal information, is the key feature of [[quantum physics]] that distinguishes it from [[classical physics]].

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

\begin{itemize}%
\item [[wave function]], [[collapse of the wave function]]

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

For comprehensive references see those at

\begin{itemize}%
\item [[quantum probability theory]]

\end{itemize}
Textbook accounts:

\begin{itemize}%
\item [[Klaas Landsman]], Sections 1.3 and 2.3 of: \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}
See also:

\begin{itemize}%
\item Quantiki, \emph{\href{https://www.quantiki.org/wiki/pure-states}{Pure states}}

\end{itemize}
Not really references on this subject, but ones referred to in the text:

\begin{itemize}%
\item [[Claude Shannon]], \emph{The lattice theory of information}, IEEE Transactions on Information Theory 1, 105--107 (1953)


\item Hua Li and Edwin Chong, \emph{On a connection between information and group lattices}, Entropy 13, 683--708 (2011) (\href{http://www.mdpi.com/1099-4300/13/3/683}{mdpi:1099-4300/13/3/683})



\end{itemize}
[[!redirects pure state]] [[!redirects pure states]]

[[!redirects mixed state]] [[!redirects mixed states]]



\end{document}