  It is one of the most important and long-standing issues of physics to derive
the irreversibility out of a time-reversal symmetric equation of motion. The
present paper considers the breaking of the time-reversal symmetry in open
quantum systems and the emergence of an arrow of time. We claim that the
time-reversal symmetric Schr\"{o}dinger equation can have eigenstates that
break the time-reversal symmetry if the system is open in the sense that it has
at least a countably infinite number of states. Such eigenstates, namely the
resonant and anti-resonant states, have complex eigenvalues. We show that,
although these states are often called "unphysical," they observe the
probability conservation in a particular way. We also comment that the
seemingly Hermitian Hamiltonian is non-Hermitian in the functional space of the
resonant and anti-resonant states, and hence there is no contradiction in the
fact that it has complex eigenvalues. We finally show how the existence of the
states that break the time-reversal symmetry affects the quantum dynamics. The
dynamics that starts from a time-reversal symmetric initial state is dominated
by the resonant states for $t>0$; this explains the phenomenon of the arrow of
time, in which the decay excels the growth. The time-reversal symmetry holds in
that the dynamics ending at a time-reversal symmetric final state is dominated
by the anti-resonant states for $t<0$.
