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\begin{document}

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\section*{trace}

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

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

[[!include monoidal categories - contents]]

\hypertarget{linear_algebra}{}\paragraph*{{Linear algebra}}\label{linear_algebra}

[[!include homotopy - contents]]

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

[[!include category theory - contents]]

\hypertarget{traces}{}\section*{{Traces}}\label{traces}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{vertical_categorification}{Vertical categorification}\dotfill \pageref*{vertical_categorification} \linebreak
\noindent\hyperlink{horizontal_categorification}{Horizontal categorification}\dotfill \pageref*{horizontal_categorification} \linebreak
\noindent\hyperlink{partial_trace}{Partial trace}\dotfill \pageref*{partial_trace} \linebreak
\noindent\hyperlink{matrix_representation}{Matrix representation}\dotfill \pageref*{matrix_representation} \linebreak
\noindent\hyperlink{example}{Example}\dotfill \pageref*{example} \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}

For $a$ a [[dualizable object]] in a [[symmetric monoidal category]] $C$ (or more generally an [[object]] in a [[traced monoidal category]]), there is a natural notion of the \emph{trace} of an [[endomorphism]] $f:a \to a$, which reproduces the ordinary notion of trace of a [[linear map]] of [[finite number|finite]] [[dimension|dimensional]] [[vector spaces]] in [[linear algebra]] for the case that $C = Vect$.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

The idea of the trace operation is easily seen in [[string diagram]] notation: essentially one takes the endomorphism $a \stackrel{f}{\to} a$, ``bends it around'' using the duality and the symmetry and connects its output to its input.

\begin{displaymath}
\itexarray{
   1 
   \\
   \;\;\;\downarrow^{tr(f)}
   \\
   1
 }
  \;\;\;
  :=
  \;\;\;
  \itexarray{
     & 1
     \\
     & \downarrow
     \\
     a^* &\otimes& a
     \\
     \downarrow^{\mathrlap{Id_{a^*}}} && \;\;\downarrow^f
     \\
     a^* &\otimes& a
     \\
     & \downarrow^{\mathrlap{b_{a^*, a}}} 
     \\
     a &\otimes& a^*
     \\
     & \downarrow
     \\
     & 1     
  }
\end{displaymath}
This definition makes sense in any [[braided monoidal category]], but often in non-symmetric cases one wants instead a slightly modified version which requires the extra structure of a [[balanced monoidal category|balancing]].

The trace of the identity $1_a:a \to a$ is called the \textbf{[[dimension]]} or [[Euler characteristic]] of $a$.

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

\begin{itemize}%
\item $C = Vect$ with its standard monoidal structure ([[tensor product]] of vector spaces): in this case tr(f) is the usual trace of a linear map;


\item $C = SuperVect = (Vect_{\mathbb{Z}_2}, \otimes, b)$, the category of $\mathbb{Z}_2$-graded vector spaces with the \emph{non}trivial symmetric braiding which is $-1$ on two odd graded vector spaces: in this case the above is the \textbf{[[supertrace]]} on supervectorspaces, $str(V) = tr(V_{even}) - tr(V_odd)$.


\item $C = Span(Top^{op})$: here the trace is the [[co-span co-trace]] which can be seen as describing the gluing of in/out boundaries of [[cobordism]]s


\item $C = Span(Grpd)$: this reproduces the notion of trace of a linear map within the interpretation of spans of groupoids as linear maps in the context of [[groupoidification]] and [[geometric function theory]], made explicit at [[span trace]]


\item In the [[symmetric monoidal (infinity,1)-category of spectra]], the trace on the identity on a [[suspension spectrum]] of a [[manifold]] $X$ is the [[Euler characteristic]] of $X$ (see there).



\end{itemize}
\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

\hypertarget{vertical_categorification}{}\subsubsection*{{Vertical categorification}}\label{vertical_categorification}

See [[trace of a category]].

\hypertarget{horizontal_categorification}{}\subsubsection*{{Horizontal categorification}}\label{horizontal_categorification}

See [[trace in a bicategory]].

\hypertarget{partial_trace}{}\subsubsection*{{Partial trace}}\label{partial_trace}

If the morphism described above is the endomorphism of a tensor product object $V \otimes W$, then there is a similarly evident way to ``bend around'' only the W-strand.

TO DO: Draw the diagram just described.

\hypertarget{matrix_representation}{}\paragraph*{{Matrix representation}}\label{matrix_representation}

Suppose $V$, $W$ are finite-dimensional vector spaces over a field, with dimensions $m$ and $n$, respectively. For any space $A$ let $L(A)$ denote the space of linear operators on $A$. The \textbf{partial trace} over $W$, Tr$_{W}$, is a mapping

\begin{displaymath}
T \in L(V \otimes W) \mapsto Tr_{W}(T) \in L(V).
\end{displaymath}
\begin{udefn}
Let $e_{1}, \ldots, e_{m}$ and $f_{1}, \ldots, f_{n}$ be bases for $V$ and $W$ respectively. Then $T$ has a matrix representation $\{a_{k l,i j}\}$ where $1 \le k,i \le m$ and $1 \le l,j \le n$ relative to the basis of the space $V \otimes W$ given by $e_{k} \otimes f_{l}$. Consider the sum

\begin{displaymath}
b_{k,i} = \sum_{j=1}^{n}a_{k j,i j}
\end{displaymath}
for $k,i$ over $1, \ldots, m$. This gives the matrix $b_{k,i}$. The associated linear operator on $V$ is independent of the choice of bases and is defined as the partial trace.

\end{udefn}
\hypertarget{example}{}\paragraph*{{Example}}\label{example}

Consider a quantum system, $\rho$, in the presence of an environment, $\rho_{env}$. Consider what is known in [[quantum information theory]] as the CNOT gate:

\begin{displaymath}
U={|00\rangle}{\langle 00|} + {|01\rangle}{\langle 01|} + {|11\rangle}{\langle 10|} + {|10\rangle}{\langle 11|}.
\end{displaymath}
Suppose our system has the simple state ${|1\rangle}{\langle 1|}$ and the environment has the simple state ${|0\rangle}{\langle 0|}$. Then $\rho \otimes \rho_{env} = {|10\rangle}{\langle 10|}$. In the quantum operation formalism we have

\begin{displaymath}
T(\rho) = \frac{1}{2}Tr_{env}U(\rho \otimes \rho_{env})U^{\dagger}
          = \frac{1}{2}Tr_{env}({|10\rangle}{\langle 10|} + {|11\rangle}{\langle 11|})
          = \frac{{|1\rangle}{\langle 1|}{\langle 0|0\rangle} + {|1\rangle}{\langle 1|}{\langle 1|1\rangle}}{2}
          = {|1\rangle}{\langle 1|}
\end{displaymath}
where we inserted the normalization factor $\frac{1}{2}$.

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

\begin{itemize}%
\item [[traced monoidal category]]


\item [[Euler characteristic]]


\item [[bicategorical trace]],

\begin{itemize}%
\item [[Reidemeister trace]]

\end{itemize}

\item [[higher trace]]


\item [[Dennis trace]], [[cyclotomic trace]]



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

The categorical notion of trace in a monoidal category is due to

\begin{itemize}%
\item [[Albrecht Dold]], and [[Dieter Puppe]], \emph{Duality, trace, and transfer} In Proceedings of the Inter- national Conference on Geometric Topology (Warsaw, 1978), pages 81\{102, Warsaw, 1980. PWN.

\end{itemize}
and

\begin{itemize}%
\item [[Max Kelly]] M. L. Laplaza, \emph{Coherence for compact closed categories} J. Pure Appl. Algebra, 19:193\{213, 1980.

\end{itemize}
Surveys include

\begin{itemize}%
\item [[Peter Selinger]], \emph{A survey of graphical languages for monoidal categories} (\href{http://www.mathstat.dal.ca/~selinger/papers/graphical.pdf}{pdf}), Section 5


\item [[Kate Ponto]], [[Mike Shulman]], \emph{Traces in symmetric monoidal categories} (\href{http://arxiv.org/pdf/1107.6032v1.pdf}{pdf}).



\end{itemize}
Generalization of this to [[indexed monoidal categories]] is in

\begin{itemize}%
\item [[Kate Ponto]], [[Mike Shulman]], \emph{Duality and traces for indexed monoidal categories}, Theory and Applications of Categories, Vol. 26, 2012, No. 23, pp 582-659 (\href{http://arxiv.org/abs/1211.1555}{arXiv:1211.1555})

\end{itemize}
and to [[bicategories]] in

\begin{itemize}%
\item [[Kate Ponto]], [[Mike Shulman]], \emph{Shadows and traces in bicategories}, JHRS, (\href{http://arxiv.org/abs/1211.1555}{arXiv:1211.1555})

\end{itemize}
Further developments are in

\begin{itemize}%
\item [[Andre Joyal]], [[Ross Street]], and [[Dominic Verity]], \emph{Traced Monoidal Categories}


\item [[David Ben-Zvi]], [[David Nadler]], \emph{Nonlinear traces} (\href{http://arxiv.org/abs/1305.7175}{arXiv:1305.7175})



\end{itemize}
For the notion of partial trace, particularly its application to [[quantum mechanics]], see:

\begin{itemize}%
\item Nielsen and Chuang, \emph{Quantum Computation and Quantum Information}

\end{itemize}
[[!redirects traces]]

[[!redirects partial trace]] [[!redirects partial traces]]



\end{document}