We apply algorithmic information theory to quantum mechanics in order to shed light on an algorithmic structure which inheres in quantum mechanics. There are two equivalent ways to define the (classical) Kolmogorov complexity K(s) of a given classical finite binary string s. In the standard way, K(s) is defined as the length of the shortest input string for the universal self-delimiting Turing machine to output s. In the other way, we first introduce the so-called universal probability m, and then define K(s) as -log_2 m(s) without using the concept of program-size. We generalize the universal probability to a matrix-valued function, and identify this function with a POVM (positive operator-valued measure). On the basis of this identification, we study a computable POVM measurement with countable measurement outcomes performed upon a finite dimensional quantum system. We show that, up to a multiplicative constant, 2^ -K(s) is the upper bound for the probability of each measurement outcome s in such a POVM measurement. In what follows, the upper bound 2^ -K(s) is shown to be optimal in a certain sense.
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 − 2 log 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.