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 June 3, 2013 10:22:04 by Urs Schreiber (82.113.98.121)