For a quantum-mechanical particle without interaction, a linear manifold of states is identified that has a preferred time direction pointing to the future. States with the broken time-reversal symmetry are singled out by their behavior under dilations. At positive times such states can be described in terms of a density operator with the property that the trace of its square decreases as the time increases. This density operator determines an entropy that approaches its least upper bound when the time tends to infinity. The expectation value of the dilation operator was negative in the distant past and will be positive in the remote future. This is the irreversible aspect of the time evolution that causes the entropy to increase, although there is no approach to equilibrium. The density operator with increasing entropy is obtained from the usual density operator by an invertible transformation that is compared with the Λ-transformation in the Prigogine theory of irreversible behavior in K-systems and large Hamiltonian systems with many resonances.