  In supervised learning, an inductive learning algorithm extracts general
rules from observed training instances, then the rules are applied to test
instances. We show that this splitting of training and application arises
naturally, in the classical setting, from a simple independence requirement
with a physical interpretation of being non-signalling. Thus, two seemingly
different definitions of inductive learning happen to coincide. This follows
from the properties of classical information that break down in the quantum
setup. We prove a quantum de Finetti theorem for quantum channels, which shows
that in the quantum case, the equivalence holds in the asymptotic setting, that
is, for large number of test instances. This reveals a natural analogy between
classical learning protocols and their quantum counterparts, justifying a
similar treatment, and allowing to inquire about standard elements in
computational learning theory, such as structural risk minimization and sample
complexity.
