nLab
trace

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.

$$\begin{array}{c}1\\ {\downarrow }^{\mathrm{tr}(f)}\\ 1\end{array}:=\begin{array}{cc}& 1\\ & \downarrow \\ a^{*}& \otimes & a\\ {\downarrow }^{{\mathrm{Id}}_{a^{*}}}& & {\downarrow }^f\\ a^{*}& \otimes & a\\ & {\downarrow }^{b_{a^{*},a}}\\ a& \otimes & a^{*}\\ & \downarrow \\ & 1\end{array}$$\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;

• $C=\mathrm{SuperVect}=({\mathrm{Vect}}_{{\mathbb{Z}}_2},\otimes ,b)$, 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.

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

$$T\in L(V\otimes W)\mapsto {\mathrm{Tr}}_W(T)\in L(V).$$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 $\{a_{kl,ij}\}$ 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

$$T(\rho )=\frac{1}{2}{\mathrm{Tr}}_{\mathrm{env}}U(\rho \otimes {\rho }_{\mathrm{env}})U^{†}=\frac{1}{2}{\mathrm{Tr}}_{\mathrm{env}}(\mid 10⟩⟨10\mid +\mid 11⟩⟨11\mid )=\frac{\mid 1⟩⟨1\mid ⟨0\mid 0⟩+\mid 1⟩⟨1\mid ⟨1\mid 1⟩}{2}=\mid 1⟩⟨1\mid$$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

• 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