  Physical laws for elementary particles can be described by the quantum
dynamics equation given a Hamiltonian. The solution are probability amplitudes
in Hilbert space that evolve over time. A probability density function over
position and time is given as the magnitude square of such probability
amplitude. An entropy can be associated with these probability densities
characterizing the position information of a particle. Coherent states are
localized wave packets and may describe the spatial distribution for some
particle states. We show that due to a dispersion property of Hamiltonians in
quantum physics, the entropy of coherent states increases over time. We
investigate a partition of the Hilbert space into four sets based on whether
the entropy is (i) increasing but not a constant, (ii) decreasing but not a
constant, (iii) a constant, (iv) oscillating.
  We propose a law that entropy (weakly) increases in time in quantum physics.
Thus, states in set (ii) are disallowed, and the states in set (iii) cannot
complete an oscillation period.
  Then, according to this law, quantum theory is not time-reversible unless the
state is in set (iii), i.e., it is a stationary state (an eigenstate of the
Hamiltonian.). This additional law in quantum theory limits physical scenarios
beyond conservation laws, providing causality reasoning by defining an arrow of
time.
