  The exponential modalities of linear logic have been used by various authors
to model infinite-dimensional quantum systems. This paper explains how these
modalities can also give rise to the complementarity principle of quantum
mechanics.
  The paper uses a formulation of quantum systems based on dagger-linear logic,
whose categorical semantics lies in mixed unitary categories, and a formulation
of measurement therein. The main result exhibits a complementary system as the
result of measurements on free exponential modalities. Recalling that, in
linear logic, exponential modalities have two distinct but dual components, !
and ?, this shows how these components under measurement become "compacted"
into the usual notion of complementary Frobenius algebras from categorical
quantum mechanics.
