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\section*{finite quantum mechanics in terms of dagger-compact categories}

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

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

[[!include physicscontents]]

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{monoidal_categories}{}\paragraph*{{Monoidal categories}}\label{monoidal_categories}

[[!include monoidal categories - contents]]

\begin{quote}%
under construction


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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{quantum_mechanical_concepts_in_compact_categories}{Quantum mechanical concepts in $\dagger$-compact categories}\dotfill \pageref*{quantum_mechanical_concepts_in_compact_categories} \linebreak
\noindent\hyperlink{classical_measurement_outcomes}{Classical measurement outcomes}\dotfill \pageref*{classical_measurement_outcomes} \linebreak
\noindent\hyperlink{complex_phases}{Complex phases}\dotfill \pageref*{complex_phases} \linebreak
\noindent\hyperlink{ComplPosMaps}{Completely positive maps}\dotfill \pageref*{ComplPosMaps} \linebreak
\noindent\hyperlink{quantum_logic}{Quantum logic}\dotfill \pageref*{quantum_logic} \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}

Central aspects of \emph{finite} [[quantum mechanics]] (with finite-dimensional [[space of states|state space]], notably for [[tensor products]] of [[qbit]] states) and [[quantum computation]] follow formally from the formal properties of the [[category]] [[Hilb|FinHilb]] of [[finite-dimensional vector space|finite-dimensional]] [[Hilbert spaces]]. These properties are axomatized by saying that [[Hilb]] is an example of a \emph{[[†-compact category]]}.

Conversely, much of finite quantum mechanics and quantum computation can be formulated in \emph{any} \dag{}-compact category, and general reasoning about \dag{}-compact categories themselves yields results about quantum mechanics and quantum computation.

A transparent [[string diagram]] calculus in \dag{}-compact categories as exposed in (\hyperlink{CoeckeKindergarten}{Coecke, Kindergarten quantum mechanics}) provides an intuitive and powerful tool for reasoning in $\dagger$-compact categories.

\hypertarget{quantum_mechanical_concepts_in_compact_categories}{}\subsection*{{Quantum mechanical concepts in $\dagger$-compact categories}}\label{quantum_mechanical_concepts_in_compact_categories}

Let $(C,\otimes,I, \dagger)$ be a [[†-compact category]].

We list various concepts in quantum mechanics and their corresponding incarnation in terms of structures in $C$.

\hypertarget{classical_measurement_outcomes}{}\subsubsection*{{Classical measurement outcomes}}\label{classical_measurement_outcomes}

An [[observable]] in quantum mechanics formulated on a [[Hilbert space]] is modeled by a [[self-adjoint operator]], and the classical measurement outcomes of this operator provide, at least under some assumptions, an [[orthogonal basis]] on the [[Hilbert space]].

That, more abstractly, the notion of orthogonal basis of an object can be phrased intrinsically inside any suitable $\dagger$-compact category is the point made in (\hyperlink{CoeckePavlovicVicary}{CoeckePavlovicVicary}).

\hypertarget{complex_phases}{}\subsubsection*{{Complex phases}}\label{complex_phases}

The underlying ``algebra of quantum amplitudes'' of the corresponding quantum mechanical system is the [[endomorphism]] [[monoid]] of the tensor unit

\begin{displaymath}
\mathbb{C}_C = End_C(I)
  \,.
\end{displaymath}
In (\hyperlink{Vicary}{Vicary}) it is shown that in $\dagger$-compact categories with all finite limits over certain ``tree-like'' diagrams compatible with the $\dagger$-structure, this $\mathbb{C}_C$ has the properties that

\begin{itemize}%
\item it is a [[field]] of characteristic 0 with involution $\dagger$;


\item the subfield $\mathbb{R}_C$ fixed under $\dagger$ is [[order]]able.



\end{itemize}
If furthermore every bounded sequence of measurements in $C$ with values in $\mathbb{R}_C$ has a least upper bound, then it follows that this field coincides with the [[complex number]]s

\begin{displaymath}
\mathbb{C}_C = \mathbb{C}
\end{displaymath}
and moreover

\begin{displaymath}
\mathbb{R}_C = \mathbb{R}
  \,.
\end{displaymath}
\hypertarget{ComplPosMaps}{}\subsubsection*{{Completely positive maps}}\label{ComplPosMaps}

The behaviour of [[quantum channel]]s and \emph{completely positive maps} has an elegant categorical description in terms of $\dagger$-compact categories. See (\hyperlink{SelingerPositive}{Selinger} and \hyperlink{CoeckePositive}{Coecke}).

\hypertarget{quantum_logic}{}\subsection*{{Quantum logic}}\label{quantum_logic}

[[symmetric monoidal categories|Symmetric monoidal categories]] such as [[†-compact categories]] have as [[internal logic]] a fragment of [[linear logic]] and as [[type theory]] a flavor of [[linear type theory]]. In this fashion everything that can be formally said about quantum mechanics in terms of [[†-compact categories]] has an equivalent expression in [[formal logic]]/[[type theory]]. It has been argued (\hyperlink{AbramskyDuncan05}{Abramsky-Duncan 05}, \hyperlink{Duncan06}{Duncan 06}) that this [[linear logic]]/[[linear type theory]] of quantum mechanics is the correct formalization of ``[[quantum logic]]''. An exposition of this point of view is in (\hyperlink{BaezStay09}{Baez-Stay 09}).

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

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


\item [[linear logic]], [[linear type theory]], [[quantum logic]], [[quantum computing]]


\item [[JBW-algebraic quantum mechanics]]


\item [[tensor network]], [[string diagram]]



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

The idea that the natural language of [[quantum mechanics]] and [[quantum computation]] is that of [[†-compact categories]] became popular with the publication

\begin{itemize}%
\item [[Samson Abramsky]] and [[Bob Coecke]], \emph{A categorical semantics of quantum protocols} , Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE Computer Science Press (2004) (\href{http://arxiv.org/abs/quant-ph/0402130}{arXiv:quant-ph/0402130})

\end{itemize}
with an expanded version in

\begin{itemize}%
\item [[Samson Abramsky]], [[Bob Coecke]], \emph{Categorical quantum mechanics}, in \emph{Handbook of Quantum Logic and Quantum Structures vol II}, Elsevier, 2008 (\href{http://arxiv.org/abs/0808.1023}{arXiv:0808.1023})

\end{itemize}
A fairly comprehensive account of the underlying theory of [[string diagrams]] is in

\begin{itemize}%
\item [[Peter Selinger]], \emph{Dagger compact closed categories and completely positive maps} (\href{http://www.mscs.dal.ca/~selinger/papers.html#dagger}{web}, \href{http://www.mscs.dal.ca/~selinger/papers/dagger.pdf}{pdf})

\end{itemize}
A pedagogical exposition of the graphical calculus is in

\begin{itemize}%
\item [[Bob Coecke]], \emph{Kindergarten quantum mechanics} (\href{http://arxiv.org/abs/quant-ph/0510032}{arXiv:quant-ph/0510032})


\item [[Bob Coecke]], \emph{Quantum Picturalism} (\href{http://arxiv.org/abs/0908.1787}{arXiv:0908.1787})



\end{itemize}
More basic introductions are in

\begin{itemize}%
\item [[Bob Coecke]], \emph{Introducing categories to the practicing physicist} (\href{http://arxiv.org/abs/0808.1032}{arXiv:0808.1032})


\item [[Bob Coecke]], \emph{Categories for the practising physicist} (\href{http://arxiv.org/abs/0905.3010}{arXiv:0905.3010})


\item [[Chris Heunen]], [[Jamie Vicary]], \emph{Lectures on categorical quantum mechanics}, 2012 (\href{https://www.cs.ox.ac.uk/files/4551/cqm-notes.pdf}{pdf})



\end{itemize}
A comprehensive collection of basics and of recent developments is in

\begin{itemize}%
\item [[Bob Coecke]], [[Ross Duncan]], \emph{Interacting Quantum Observables: Categorical Algebra and Diagrammatics} (\href{http://arxiv.org/abs/0906.4725}{arXiv:0906.4725})

\end{itemize}
The formalization of orthogonal bases in $\dagger$-compact categories is in

\begin{itemize}%
\item [[Bob Coecke]], [[Dusko Pavlovic]], [[Jamie Vicary]], \emph{A new description of orthogonal bases} (\href{http://arxiv.org/abs/0810.0812}{arXiv:0810.0812})

\end{itemize}
The role of [[complex numbers]] in general $\dagger$-compact categories is discussed in

\begin{itemize}%
\item [[Jamie Vicary]], \emph{Completeness of $\dagger$-categories and the complex numbers} (\href{http://arxiv.org/abs/0807.2927}{arXiv:0807.2927})

\end{itemize}
[[quantum operation|Completely positive maps]] in terms of $\dagger$-categories are discussed in

\begin{itemize}%
\item [[Peter Selinger]], \emph{Dagger compact closed categories and completely positive maps} (\href{http://www.mscs.dal.ca/~selinger/papers.html#dagger}{web}, \href{http://www.mscs.dal.ca/~selinger/papers/dagger.pdf}{pdf})


\item [[Bob Coecke]], \emph{Complete positivity without compactness} (\href{http://www.comlab.ox.ac.uk/files/666/RR-07-05.pdf}{pdf})



\end{itemize}
The relation to [[quantum logic]]/[[linear logic]] has been expolred in

\begin{itemize}%
\item [[Samson Abramsky]], [[Ross Duncan]], \emph{A Categorical Quantum Logic} (\href{http://arxiv.org/abs/quant-ph/0512114}{arXiv:quant-ph/0512114})


\item [[Ross Duncan]], \emph{Types for quantum mechanics}, 2006 (\href{http://homepages.ulb.ac.be/~rduncan/papers/rduncan-thesis.pdf}{pdf}, \href{http://www.cs.ox.ac.uk/people/ross.duncan/talks/2005/pps-22-05-2005.pdf}{slides})



\end{itemize}
An exposition along these lines is in

\begin{itemize}%
\item [[John Baez]], [[Mike Stay]], \emph{Physics, topology, logic and computation: a rosetta stone}, \href{http://arxiv.org/abs/0903.0340}{arxiv/0903.0340}; in ``New Structures for Physics'', ed. Bob Coecke, Lecture Notes in Physics \textbf{813}, Springer, Berlin, 2011, pp. 95-174

\end{itemize}
[[!redirects finite quantum mechanics in terms of †-compact categories]]

[[!redirects quantum mechanics in terms of †-compact categories]] [[!redirects quantum mechanics in terms of dagger-compact categories]]



\end{document}