  Motivated by studies of typical properties of quantum states in statistical
mechanics, we introduce phase-random states, an ensemble of pure states with
fixed amplitudes and uniformly distributed phases in a fixed basis. We first
show that canonical states typically appear in subsystems of phase-random
states. We then investigate the simulatability of phase-random states, which is
directly related to that of time evolution in closed systems, by studying their
entanglement properties. We find that starting from a separable state, time
evolutions under Hamiltonians composed of only separable eigenstates generate
extremely high entanglement and are difficult to simulate with matrix product
states. We also show that random quantum circuits consisting of only two-qubit
diagonal unitaries can generate an ensemble with the same average entanglement
as phase-random states.
