# nLab trace

### Context

#### Monoidal categories

monoidal categories

## With traces

• trace

• traced monoidal category?

category theory

# Traces

## Definition

If $a$ is a dualizable object in a symmetric monoidal category $C$, there is a notion of the trace of an endomorphism $f:a\to a$, which reproduces the ordinary notion of trace of a linear map of finite dimensional vector spaces for the case that $C=\mathrm{Vect}$.

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.

$\begin{array}{c}1\\ \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{↓}^{\mathrm{tr}\left(f\right)}\\ 1\end{array}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}:=\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\begin{array}{cc}& 1\\ & ↓\\ {a}^{*}& \otimes & a\\ {↓}^{{\mathrm{Id}}_{{a}^{*}}}& & \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{↓}^{f}\\ {a}^{*}& \otimes & a\\ & {↓}^{{b}_{{a}^{*},a}}\\ a& \otimes & {a}^{*}\\ & ↓\\ & 1\end{array}$\array{ 1 \\ \;\;\;\downarrow^{tr(f)} \\ 1 } \;\;\; := \;\;\; \array{ & 1 \\ & \downarrow \\ a^* &\otimes& a \\ \downarrow^{\mathrlap{Id_{a^*}}} && \;\;\downarrow^f \\ a^* &\otimes& a \\ & \downarrow^{\mathrlap{b_{a^*, a}}} \\ a &\otimes& a^* \\ & \downarrow \\ & 1 }

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 balancing.

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

## Examples

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

• $C=\mathrm{SuperVect}=\left({\mathrm{Vect}}_{{ℤ}_{2}},\otimes ,b\right)$, the category of ${ℤ}_{2}$-graded vector spaces with the nontrivial symmetric braiding which is $-1$ on two odd graded vector spaces: in this case the above is the supertrace on supervectorspaces, $\mathrm{str}\left(V\right)=\mathrm{tr}\left({V}_{\mathrm{even}}\right)-\mathrm{tr}\left({V}_{\mathrm{odd}}\right)$.

• $C=\mathrm{Span}\left({\mathrm{Top}}^{\mathrm{op}}\right)$: here the trace is the co-span co-trace which can be seen as describing the gluing of in/out boundaries of cobordisms

• $C=\mathrm{Span}\left(\mathrm{Grpd}\right)$: 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

## Generalizations

### 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.

#### Matrix representation

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

$T\in L\left(V\otimes W\right)↦{\mathrm{Tr}}_{W}\left(T\right)\in L\left(V\right).$T \in L(V \otimes W) \mapsto Tr_{W}(T) \in L(V).
###### Definition

Let ${e}_{1},\dots ,{e}_{m}$ and ${f}_{1},\dots ,{f}_{n}$ be bases for $V$ and $W$ respectively. Then $T$ has a matrix representation $\left\{{a}_{kl,ij}\right\}$ 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

${b}_{k,i}=\sum _{j=1}^{n}{a}_{kj,ij}$b_{k,i} = \sum_{j=1}^{n}a_{k j,i j}

for $k,i$ over $1,\dots ,m$. 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.

#### Example

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

$U=\mid 00⟩⟨00\mid +\mid 01⟩⟨01\mid +\mid 11⟩⟨10\mid +\mid 10⟩⟨11\mid .$U={|00\rangle}{\langle 00|} + {|01\rangle}{\langle 01|} + {|11\rangle}{\langle 10|} + {|10\rangle}{\langle 11|}.

Suppose our system has the simple state $\mid 1⟩⟨1\mid$ and the environment has the simple state $\mid 0⟩⟨0\mid$. Then $\rho \otimes {\rho }_{\mathrm{env}}=\mid 10⟩⟨10\mid$. In the quantum operation formalism we have

$T\left(\rho \right)=\frac{1}{2}{\mathrm{Tr}}_{\mathrm{env}}U\left(\rho \otimes {\rho }_{\mathrm{env}}\right){U}^{†}=\frac{1}{2}{\mathrm{Tr}}_{\mathrm{env}}\left(\mid 10⟩⟨10\mid +\mid 11⟩⟨11\mid \right)=\frac{\mid 1⟩⟨1\mid ⟨0\mid 0⟩+\mid 1⟩⟨1\mid ⟨1\mid 1⟩}{2}=\mid 1⟩⟨1\mid$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|}

where we inserted the normalization factor $\frac{1}{2}$.

## References

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

• Albrecht Dold, and Dieter Puppe, Duality, trace, and transfer In Proceedings of the Inter- national Conference on Geometric Topology (Warsaw, 1978), pages 81{102, Warsaw, 1980. PWN.

and

• Max Kelly M. L. Laplaza, Coherence for compact closed categories J. Pure Appl. Algebra, 19:193{213, 1980.

A survey is in

Further developments are in

For partial trace, particularly its application to quantum mechanics, see:

• Nielsen and Chuang, Quantum Computation and Quantum Information

Revised on May 18, 2013 19:17:21 by Anonymous Coward (74.69.73.189)