nLab trace

Traces

Context

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

category theory

Traces

Idea

Given an endomorphism $f:A\to A$ in a traced monoidal category $\mathscr{C}$ there is a natural notation of a trace $\tr(f):1\to 1$, where $1$ is the tensor unit. When $\mathscr{C}$ is a $\mathbb{C}$-linear category for which the tensor unit (e.g. a fusion categories), we can canonically identity $\tr(f)$ with a complex number. That is, the trace of $f$ is the unique scalar $\lambda$ such that

$\tr(f)=\lambda \cdot \text{id}_A.$

In the case that $f$ is a linear endomorphism of a finite dimensional vector space, we recover the usual notion of trace from linear algebra.

Definition

The idea of the trace operation is easily seen in string diagram notation: essentially one takes an endomorphism $f: A\xrightarrow{}A$, and “closes the loop”.

$\array{ 1 \\ \;\;\;\downarrow^{tr(f)} \\ 1 } \;\;\; := \;\;\; \array{ & 1 \\ & \downarrow \\ a^* &\otimes& a \\ \downarrow^{\mathrlap{Id_{a^*}}} && \;\;\downarrow^f \\ a^* &\otimes& a \\ & \downarrow^{\mathrlap{b_{a^*, a}}} \\ a &\otimes& a^* \\ & \downarrow \\ & 1 }$

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

• $C = Vect_k$ with its standard monoidal structure (tensor product of vector spaces): in this case $tr(f)$ is the usual trace of a linear map. Indeed, suppose $f$ is an endomorphism on a finite dimensional $k$-vector space $V$. Then, the trace $tr(f) \colon k\to k$ is given by the following composition.

$k\xrightarrow{\eta} V\otimes V^\ast\xrightarrow{f\otimes id}V\otimes V^\ast\xrightarrow{\varepsilon} k$

Here $\varepsilon$ is the standard evaluation-pairing. The map $\eta$ is obtained from the dual linear map of $\varepsilon$ by applying the following isomorphisms, see (Dold & Puppe 1978).

$k\cong k^\ast,\qquad (V\otimes V^\ast)^\ast\cong(V^\ast)^\ast\otimes V^\ast\cong V\otimes V^\ast$

One can then compute that

$\eta(\lambda)=\sum_{i=1}^n\lambda (e_i\otimes e_i^\ast),$

where $\{e_1,\ldots,e_n\}$ is a linear basis of $V$ and $\{e_1^\ast,\ldots,e_n^\ast\}$ is the dual basis. Now, if $f$ is given in this basis by a matrix $(a_ij)$, then

$(f\otimes id)(\eta(\lambda))=\lambda\sum\begin{pmatrix}a_{1i}\\\vdots\\a_{ni}\end{pmatrix}\otimes e_i^\ast.$

Finally, $(tr(f))(\lambda)$ is obtained by applying the pairing on this tensor: $(tr(f))(\lambda)=\lambda\sum a_{ii}$.

• $C = SuperVect = (Vect_{\mathbb{Z}_2}, \otimes, b)$, the category of $\mathbb{Z}_2$-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, $str(V) = tr(V_{even}) - tr(V_odd)$.

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

• $C = Span(Grpd)$: 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.

• 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).

Generalizations

Horizontal categorification

See trace in a bicategory?.

Partial trace

Let $V \in C$ be a dualizable object, and $W$ any object. From an endomorphism $f$ of $V \otimes W$, one can produce an endomorphism $tr_V(f)$ of $W$, by applying the duality data of $V$. In terms of string diagrams, this is “bending along” the strand representing $V$.

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 $L(A)$ denote the space of linear operators on $A$. The partial trace over $W$, Tr$_{W}$, is a mapping

$T \in L(V \otimes W) \mapsto Tr_{W}(T) \in L(V).$
Definition

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

$b_{k,i} = \sum_{j=1}^{n}a_{k j,i j}$

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.

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:

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

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

$T(\rho) = \frac{1}{2}Tr_{env}U(\rho \otimes \rho_{env})U^{\dagger} = \frac{1}{2}Tr_{env}({|10\rangle}{\langle 10|} + {|11\rangle}{\langle 11|}) = \frac{{|1\rangle}{\langle 1|}{\langle 0|0\rangle} + {|1\rangle}{\langle 1|}{\langle 1|1\rangle}}{2} = {|1\rangle}{\langle 1|}$

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.

Surveys include

Generalization of this to indexed monoidal categories is in

and to bicategories in

Further developments are in

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

• Nielsen and Chuang, Quantum Computation and Quantum Information

Last revised on November 2, 2023 at 04:01:47. See the history of this page for a list of all contributions to it.