  Machine learning, one of today's most rapidly growing interdisciplinary
fields, promises an unprecedented perspective for solving intricate quantum
many-body problems. Understanding the physical aspects of the representative
artificial neural-network states is recently becoming highly desirable in the
applications of machine learning techniques to quantum many-body physics. Here,
we study the quantum entanglement properties of neural-network states, with a
focus on the restricted-Boltzmann-machine (RBM) architecture. We prove that the
entanglement of all short-range RBM states satisfies an area law for arbitrary
dimensions and bipartition geometry. For long-range RBM states we show by using
an exact construction that such states could exhibit volume-law entanglement,
implying a notable capability of RBM in representing efficiently quantum states
with massive entanglement. We further examine generic RBM states with random
weight parameters. We find that their averaged entanglement entropy obeys
volume-law scaling and meantime strongly deviates from the Page-entropy of the
completely random pure states. We show that their entanglement spectrum has no
universal part associated with random matrix theory and bears a Poisson-type
level statistics. Using reinforcement learning, we demonstrate that RBM is
capable of finding the ground state (with power-law entanglement) of a model
Hamiltonian with long-range interaction. In addition, we show, through a
concrete example of the one-dimensional symmetry-protected topological cluster
states, that the RBM representation may also be used as a tool to analytically
compute the entanglement spectrum. Our results uncover the unparalleled power
of artificial neural networks in representing quantum many-body states, which
paves a novel way to bridge computer science based machine learning techniques
to outstanding quantum condensed matter physics problems.
