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

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 $\frac{1}{2}$.

References

• Andre Joyal, Ross Street, and Dominic Verity, Traced Monoidal Categories
• Dold, Albrecht and Puppe, Dieter, Duality, trace, and transfer
• 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