nLab partial trace




A partial trace is a trace applied to (only) one factor in a tensor product.

The partial trace is generalized by traced monoidal categories. That is, a traced monoidal category is a monoidal category paired with an operation which behaves like the partial trace.


Let Vec k\text{Vec}_k denote the category of finite-dimensional vector spaces over a field kk. The partial trace gives a compatible collection of linear maps

tr W:End(VW)End(V)\tr_W \colon \text{End}(V\otimes W)\to \text{End}(V)

given by

tr W(fg)=tr(g)f, \tr_W(f\otimes g) \,=\, \tr(g)\cdot f \,,

where “trtr” on the right denotes the usual trace.

Explicitly, the partial trace can also be defined as follows:

Let f:VWVWf \colon V\otimes W\to V\otimes W be an endomorphism. Let v 1,,v mv_{1}, \ldots, v_{m} and w 1,,w nw_{1}, \ldots, w_{n} be linear bases for VV and WW respectively. Then ff has a matrix representation {a kl,ij}\{a_{kl,ij}\} 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 corresponds to the partial trace.

String diagram representations

Given vector spaces A,BA,B, or more generally, given objects A,BA,B in a traced monoidal category, we will often write the partial trace using string diagrams as follows:

Considering Vec k\text{Vec}_k as a pivotal category, this diagram is completely rigorous. To make sure this notation is consistent, we verify

Additionally the closed loop can only be trace. This is because the axioms of a pivotal category we can move around the loop, so

It is a standard fact from linear algebra that the trace is uniquely characterized by the fact that it is linear and acts the same on fgf\circ g and gfg\circ f for all f,g:AAf,g:A\to A.

Last revised on September 24, 2023 at 09:56:40. See the history of this page for a list of all contributions to it.