nLab
trace

Context

Monoidal categories

Linear algebra

Category theory

Traces

Idea

For aa a dualizable object in a symmetric monoidal category CC (or more generally an object in a traced monoidal category), there is a natural notion of the trace of an endomorphism f:aaf:a \to a, which reproduces the ordinary notion of trace of a linear map of finite dimensional vector spaces in linear algebra for the case that C=VectC = Vect.

Definition

The idea of the trace operation is easily seen in string diagram notation: essentially one takes the endomorphism afaa \stackrel{f}{\to} a, “bends it around” using the duality and the symmetry and connects its output to its input.

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.

Examples

  • C=VectC = 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=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).

Generalizations

Categorification

See trace of a category

Partial trace

If the morphism described above is the endomorphism of a tensor product object VWV \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 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).
Definition

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.

Example

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}.

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

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

Revised on September 6, 2014 12:36:59 by Urs Schreiber (185.26.182.28)