  Motivated by the great success of classical generative models in machine
learning, enthusiastic exploration of their quantum version has recently
started. To depart on this journey, it is important to develop a relevant
metric to evaluate the quality of quantum generative models; in the classical
case, one such example is the (classical) inception score (cIS). In this paper,
as a natural extension of cIS, we propose the quantum inception score (qIS) for
quantum generators. Importantly, qIS relates the quality to the Holevo
information of the quantum channel that classifies a given dataset. In this
context, we show several properties of qIS. First, qIS is greater than or equal
to the corresponding cIS, which is defined through projection measurements on
the system output. Second, the difference between qIS and cIS arises from the
presence of quantum coherence, as characterized by the resource theory of
asymmetry. Third, when a set of entangled generators is prepared, there exists
a classifying process leading to the further enhancement of qIS. Fourth, we
harness the quantum fluctuation theorem to characterize the physical limitation
of qIS. Finally, we apply qIS to assess the quality of the one-dimensional spin
chain model as a quantum generative model, with the quantum convolutional
neural network as a quantum classifier, for the phase classification problem in
the quantum many-body physics.
