  Quantum classification and hypothesis testing are two tightly related
subjects, the main difference being that the former is data driven: how to
assign to quantum states $\rho(x)$ the corresponding class $c$ (or hypothesis)
is learnt from examples during training, where $x$ can be either tunable
experimental parameters or classical data "embedded" into quantum states. Does
the model generalize? This is the main question in any data-driven strategy,
namely the ability to predict the correct class even of previously unseen
states. Here we establish a link between quantum machine learning
classification and quantum hypothesis testing (state and channel
discrimination) and then show that the accuracy and generalization capability
of quantum classifiers depend on the (R\'enyi) mutual informations $I(C{:}Q)$
and $I_2(X{:}Q)$ between the quantum state space $Q$ and the classical
parameter space $X$ or class space $C$. Based on the above characterization, we
then show how different properties of $Q$ affect classification accuracy and
generalization, such as the dimension of the Hilbert space, the amount of
noise, and the amount of neglected information from $X$ via, e.g., pooling
layers. Moreover, we introduce a quantum version of the Information Bottleneck
principle that allows us to explore the various tradeoffs between accuracy and
generalization. Finally, in order to check our theoretical predictions, we
study the classification of the quantum phases of an Ising spin chain, and we
propose the Variational Quantum Information Bottleneck (VQIB) method to
optimize quantum embeddings of classical data to favor generalization.
