  We consider the continuous quantum measurement of a two-level system, for
example, a single-Cooper-pair box measured by a single-electron transistor or a
double-quantum dot measured by a quantum point contact. While the approach most
commonly used describes the gradual decoherence of the system due to the
measurement, we show that when taking into account the detector output, we get
the opposite effect: gradual purification of the density matrix. The
competition between purification due to measurement and decoherence due to
interaction with the environment can be described by a simple Langevin equation
which couples the random evolution of the system density matrix and the
stochastic detector output. The gradual density matrix purification due to
continuous measurement may be verified experimentally using present-day
technology. The effect can be useful for quantum computing.
