  In this paper we give a definition for the Kolmogorov complexity of a pure
quantum state. In classical information theory the algorithmic complexity of a
string is a measure of the information needed by a universal machine to
reproduce the string itself. We define the complexity of a quantum state by
means of the classical description complexity of an (abstract) experimental
procedure that allows us to prepare the state with a given fidelity. We argue
that our definition satisfies the intuitive idea of complexity as a measure of
``how difficult'' it is to prepare a state. We apply this definition to give an
upper bound on the algorithmic complexity of a number of states.
