  When a quantum pure state is drawn uniformly at random from a Hilbert space,
the state is typically highly entangled. This property of a random state is
known as generic entanglement of quantum states and has been long investigated
from many perspectives, ranging from the black hole science to quantum
information science. In this paper, we address the question of how symmetry of
quantum states changes the properties of generic entanglement. More
specifically, we study bipartite entanglement entropy of a quantum state that
is drawn uniformly at random from an invariant subspace of a given symmetry. We
first extend the well-known concentration formula to the one applicable to any
subspace and then show that 1. quantum states in the subspaces associated with
an axial symmetry are still highly entangled, though it is less than that of
the quantum states without symmetry, 2. quantum states associated with the
permutation symmetry are significantly less entangled, and 3. quantum states
with translation symmetry are as entangled as the generic one. We also
numerically investigate the phase-transition behavior of the distribution of
generic entanglement, which indicates that the phase transition seems to still
exist even when random states have symmetry.
