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

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”.

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

Vertical categorification

See trace of a category.

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 $V$.

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

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:

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

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

Related concepts

• relation between trace and determinant

• traced monoidal category

• Euler characteristic

• bicategorical trace,

  • Reidemeister trace

• higher trace

• Dennis trace, cyclotomic trace

• trace of horizontal to round chord diagrams

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

• Peter Selinger, A survey of graphical languages for monoidal categories (pdf), Section 5

• Kate Ponto, Mike Shulman, Traces in symmetric monoidal categories (pdf).

Generalization of this to indexed monoidal categories is in

• Kate Ponto, Mike Shulman, Duality and traces for indexed monoidal categories, Theory and Applications of Categories, Vol. 26, 2012, No. 23, pp 582-659 (arXiv:1211.1555)

and to bicategories in

• Kate Ponto, Mike Shulman, Shadows and traces in bicategories, JHRS, (arXiv:0910.1306)

Further developments are in

• Andre Joyal, Ross Street, and Dominic Verity, Traced Monoidal Categories

• David Ben-Zvi, David Nadler, Nonlinear traces (arXiv:1305.7175)

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

• Nielsen and Chuang, Quantum Computation and Quantum Information