Quantum algorithmic entropy

Abstract

Extends algorithmic information theory to quantum mechanics, taking a universal semi-computable density matrix ("universal probability") as a starting point, and defines 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 P.M.B. Vita/spl acute/nyi (1999) (we show that the latter is sometimes as large as 2n-2 log n) and the qubit complexity defined by A. Berthiaume et al. (2000). The "cloning" properties of our complexity measure are similar to those of qubit complexity.