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\begin{document}

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\section*{quantum information}

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

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

[[!include physicscontents]]

\begin{quote}%
under construction


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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{quantum_protocols_and_algorithms}{Quantum protocols and algorithms}\dotfill \pageref*{quantum_protocols_and_algorithms} \linebreak
\noindent\hyperlink{categorical_formulation}{Categorical formulation}\dotfill \pageref*{categorical_formulation} \linebreak
\noindent\hyperlink{graphical_notation}{Graphical notation}\dotfill \pageref*{graphical_notation} \linebreak
\noindent\hyperlink{extensions}{Extensions}\dotfill \pageref*{extensions} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{related_concepts_in_lab}{Related concepts in $n$Lab}\dotfill \pageref*{related_concepts_in_lab} \linebreak
\noindent\hyperlink{books}{Books}\dotfill \pageref*{books} \linebreak
\noindent\hyperlink{articles}{Articles}\dotfill \pageref*{articles} \linebreak


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

Quantum information refers to data that can be physically stored in a [[quantum mechanics|quantum system]].

Quantum information theory is the study of how such information can be encoded, measured, and manipulated. A notable sub-field is \emph{quantum computation}, a term often used synonymously with quantum information theory, which studies protocols and algorithms that use quantum systems to perform computations.

Categorical quantum information refers to a program in which the cogent aspects of [[Hilbert space]]-based quantum information theory are abstracted to the level of [[symmetric monoidal categories]].

\hypertarget{quantum_protocols_and_algorithms}{}\subsection*{{Quantum protocols and algorithms}}\label{quantum_protocols_and_algorithms}

Brief synopsis of teleportation, [[entanglement]] swapping, BB84, E91, Deutsch-Jozsa, Shor should go here\ldots{}

\hypertarget{categorical_formulation}{}\subsection*{{Categorical formulation}}\label{categorical_formulation}

There is a formulation of (aspects of) [[quantum mechanics in terms of dagger-compact categories]]. This lends itself to (and is in fact motivated by) to a discussion of quantum information.

The linear adjoint $(-)^\dagger$ gives Hilbert spaces the structure of a [[†-category]]. The category of Hilb of Hilbert spaces forms a \textbf{\dag{}-symmetric monoidal category}, that is, a [[symmetric monoidal category]] equipped with a symmetric monoidal functor $(-)^\dagger$ from $Hilb^{op}$ to $Hilb$. Furthermore, the category FHilb of \emph{finite dimensional} Hilbert spaces forms a \textbf{[[dagger-compact category|†-compact closed category]]}, or a [[compact closed category]] such that $A_*$ := $(A^*)^\dagger = (A^\dagger)^*$ and $(\eta_A)^\dagger = \epsilon_{A^*}$.

\hypertarget{graphical_notation}{}\subsubsection*{{Graphical notation}}\label{graphical_notation}

[[Aleks Kissinger]]: Much of this could probably be incorporated into the page on [[string diagrams]].

Morphisms in a monoidal category (and 2-categories in general) are inherently two dimensional, where $\circ$ is \emph{vertical} composition and $\otimes$ is \emph{horizontal} composition. These satisfy an interchange law:

\begin{displaymath}
(f_1 \otimes f_2) \circ (g_1 \otimes g_2) = (f_1 \circ g_1) \otimes (f_2 \circ g_2)
\end{displaymath}
So, if we think of these four morphisms as occupying a spot in 2 dimensional space:

[[Aleks Kissinger]]: TODO: figure

we realize that the bracketing from above is essentially meaningless syntax. This notion is the guiding concept for the graphical notation of monoidal categories, or string diagrams. In this notation, we represent objects $A,B$ as directed strings and arrows $f : A \rightarrow B$ as boxes.

We represent the tensor product as juxtaposition:

and composition as graph composition:

That is, we perform a pushout along the common edge in the category of typed graphs with boundaries. Consider the interchange law from above, but replacing some of the arrows with identities.

\begin{displaymath}
(f \otimes 1_D) \circ (1_A \otimes g) = (f \circ 1_A) \otimes (1_D \circ g) =
(1_B \circ f) \otimes (g \circ 1_C) = (1_B \otimes g) \circ (f \otimes 1_C)
\end{displaymath}
Graphically, this means we can ``slide boxes'' past each other.

\hypertarget{extensions}{}\subsection*{{Extensions}}\label{extensions}

CPM, classical structures, \ldots{}

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

\begin{itemize}%
\item wikipedia \href{https://en.wikipedia.org/wiki/Quantum_information}{quantum information}, \href{https://en.wikipedia.org/wiki/Bures_metric}{Bures metric}

\end{itemize}
\hypertarget{related_concepts_in_lab}{}\paragraph*{{Related concepts in $n$Lab}}\label{related_concepts_in_lab}

\begin{itemize}%
\item [[quantum logic]], [[quantum theory]], [[Kochen-Specker theorem]]


\item [[quantum computation]], [[qbit]]


\item [[information theory]], [[information geometry]]



\end{itemize}
\hypertarget{books}{}\paragraph*{{Books}}\label{books}

\begin{itemize}%
\item Masahito Hayashi, \emph{Quantum information theory - mathematical foundation} Graduate Texts in Physics (2017)
\item Nielsen, Chuang, \emph{Quantum computation and quantum information}, Cambridge Univeristy Press, \href{https://ncatlab.org/nlab/files/NielsenChuangQuantumComputation.pdf}{ch1 pdf}

\end{itemize}
\hypertarget{articles}{}\paragraph*{{Articles}}\label{articles}

\begin{itemize}%
\item Carmen Maria Constantin, \emph{Sheaf-theoretic methods in quantum mechanics and quantum information theory}, PhD thesis, Oxford 2015 \href{https://arxiv.org/abs/1510.02561}{arxiv/1510.02561}
\item Samson Abramsky, Adam Brandenburger, \emph{The sheaf-theoretic structure of nonlocality and contextuality}, \href{https://arxiv.org/abs/1102.0264}{arxiv/1102.0264}
\item Dominik Šafránek, \emph{Simple expression for the quantum Fisher information matrix), Phys. Rev. A97 (2018) \href{https://doi.org/10.1103/PhysRevA.97.042322}{doi}}
\item Roman Orus, \emph{Entanglement, quantum phase transitions and quantum algorithms} arxiv:\href{https://arxiv.org/abs/quant-ph/0608013}{quant-ph/0608013}

\end{itemize}
\begin{quote}%
In Chapter 1 we consider the irreversibility of renormalization group flows from a quantum information perspective by using majorization theory and conformal field theory.


\end{quote}
[[!redirects quantum information theory]]



\end{document}