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.

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;

• $C=\mathrm{SuperVect}=({\mathrm{Vect}}_{{\mathbb{Z}}_2},\otimes ,b)$, 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, $\mathrm{str}(V)=\mathrm{tr}(V_{\mathrm{even}})-\mathrm{tr}(V_{\mathrm{odd}})$.

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

• $C=\mathrm{Span}(\mathrm{Grpd})$: 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

Generalizations

Categorification

See trace of a category

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.

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, ${\rho }_{\mathrm{env}}$. Consider what is known in quantum information theory as the CNOT gate:

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

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

Related concepts

• Euler characteristic

• bicategorical trace,

  • Reidemeister trace

• higher trace

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

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

Further developments are in

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

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

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

• Nielsen and Chuang, Quantum Computation and Quantum Information