  We extend algorithmic information theory to quantum mechanics, taking a
universal semicomputable density matrix (``universal probability'') as a
starting point, and define complexity (an operator) as its negative logarithm.
  A number of properties of Kolmogorov complexity extend naturally to the new
domain. Approximately, a quantum state is simple if it is within a small
distance from a low-dimensional subspace of low Kolmogorov complexity. The von
Neumann entropy of a computable density matrix is within an additive constant
from the average complexity. Some of the theory of randomness translates to the
new domain.
  We explore the relations of the new quantity to the quantum Kolmogorov
complexity defined by Vitanyi (we show that the latter is sometimes as large as
2n - 2log n and the qubit complexity defined by Berthiaume, Dam and Laplante.
The ``cloning'' properties of our complexity measure are similar to those of
qubit complexity.
