nLab trace



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Given an endomorphism f:AAf:A\to A in a traced monoidal category 𝒞\mathscr{C} there is a natural notation of a trace tr(f):11\tr(f):1\to 1, where 11 is the tensor unit. When 𝒞\mathscr{C} is a \mathbb{C}-linear category for which the tensor unit (e.g. a fusion categories), we can canonically identity tr(f)\tr(f) with a complex number. That is, the trace of ff is the unique scalar λ\lambda such that

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

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


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

1 tr(f) 1:= 1 a * a Id a * f a * a b a *,a a a * 1 \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:aa1_a:a \to a is called the dimension or Euler characteristic of aa.


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

    kηVV *fidVV *εkk\xrightarrow{\eta} V\otimes V^\ast\xrightarrow{f\otimes id}V\otimes V^\ast\xrightarrow{\varepsilon} k

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

    kk *,(VV *) *(V *) *V *VV *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

    η(λ)= i=1 nλ(e ie i *),\eta(\lambda)=\sum_{i=1}^n\lambda (e_i\otimes e_i^\ast),

    where {e 1,,e n}\{e_1,\ldots,e_n\} is a linear basis of VV and {e 1 *,,e n *}\{e_1^\ast,\ldots,e_n^\ast\} is the dual basis. Now, if ff is given in this basis by a matrix (a ij)(a_ij), then

    (fid)(η(λ))=λ(a 1i a ni)e i *.(f\otimes id)(\eta(\lambda))=\lambda\sum\begin{pmatrix}a_{1i}\\\vdots\\a_{ni}\end{pmatrix}\otimes e_i^\ast.

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

  • C=SuperVect=(Vect 2,,b)C = SuperVect = (Vect_{\mathbb{Z}_2}, \otimes, b), the category of 2\mathbb{Z}_2-graded vector spaces with the nontrivial symmetric braiding which is 1-1 on two odd graded vector spaces: in this case the above is the supertrace on supervectorspaces, str(V)=tr(V even)tr(V odd)str(V) = tr(V_{even}) - tr(V_odd).

  • C=Span(Top op)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)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 XX is the Euler characteristic of XX (see there).


Vertical categorification

See trace of a category.

Horizontal categorification

See trace in a bicategory?.

Partial trace

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

TO DO: Draw the diagram just described.

Matrix representation

Suppose VV, WW are finite-dimensional vector spaces over a field, with dimensions mm and nn, respectively. For any space AA let L(A)L(A) denote the space of linear operators on AA. The partial trace over WW, TrW_{W}, is a mapping

TL(VW)Tr W(T)L(V). T \in L(V \otimes W) \mapsto Tr_{W}(T) \in L(V).

Let e 1,,e me_{1}, \ldots, e_{m} and f 1,,f nf_{1}, \ldots, f_{n} be bases for VV and WW respectively. Then TT has a matrix representation {a kl,ij}\{a_{k l,i j}\} where 1k,im1 \le k,i \le m and 1l,jn1 \le l,j \le n relative to the basis of the space VWV \otimes W given by e kf le_{k} \otimes f_{l}. Consider the sum

b k,i= j=1 na kj,ij b_{k,i} = \sum_{j=1}^{n}a_{k j,i j}

for k,ik,i over 1,,m1, \ldots, m. This gives the matrix b k,ib_{k,i}. The associated linear operator on VV is independent of the choice of bases and is defined as the partial trace.


Consider a quantum system, ρ\rho, in the presence of an environment, ρ env\rho_{env}. Consider what is known in quantum information theory as the CNOT gate:

U=|0000|+|0101|+|1110|+|1011|. U={|00\rangle}{\langle 00|} + {|01\rangle}{\langle 01|} + {|11\rangle}{\langle 10|} + {|10\rangle}{\langle 11|}.

Suppose our system has the simple state |11|{|1\rangle}{\langle 1|} and the environment has the simple state |00|{|0\rangle}{\langle 0|}. Then ρρ env=|1010|\rho \otimes \rho_{env} = {|10\rangle}{\langle 10|}. In the quantum operation formalism we have

T(ρ)=12Tr envU(ρρ env)U =12Tr env(|1010|+|1111|)=|11|0|0+|11|1|12=|11| 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 12\frac{1}{2}.


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.


  • 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.