  We develop a theory of the algorithmic information in bits contained in an
individual pure quantum state. This extends classical Kolmogorov complexity to
the quantum domain retaining classical descriptions. Quantum Kolmogorov
complexity coincides with the classical Kolmogorov complexity on the classical
domain. Quantum Kolmogorov complexity is upper bounded and can be effectively
approximated from above under certain conditions. With high probability a
quantum object is incompressible. Upper- and lower bounds of the quantum
complexity of multiple copies of individual pure quantum states are derived and
may shed some light on the no-cloning properties of quantum states. In the
quantum situation complexity is not sub-additive. We discuss some relations
with ``no-cloning'' and ``approximate cloning'' properties.
