  We consider a notion of complexity of quantum channels in relativistic
continuum quantum field theory (QFT) defined by the distance to the trivial
(identity) channel. Our distance measure is based on a specific divergence
between quantum channels derived from the Belavkin-Staszewski (BS) divergence.
We prove in the prerequisite generality necessary for the algebras in QFT that
the corresponding complexity has several reasonable properties: (i) the
complexity of a composite channel is not larger than the sum of its parts, (ii)
it is additive for channels localized in spacelike separated regions, (iii) it
is convex, (iv) for an $N$-ary measurement channel it is $\log N$, (v) for a
conditional expectation associated with an inclusion of QFTs with finite Jones
index it is given by $\log (\text{Jones Index})$. The main technical tool in
our work is a new variational principle for the BS divergence.
