Quantum Kolmogorov Complexity

Abstract

In this paper we give a definition for quantum Kolmogorov complexity. In the classical setting, the Kolmogorov complexity of a string is the length of the shortest program that can produce this string as its output. It is a measure of the amount of innate randomness (or information) contained in the string. We define the quantum Kolmogorov complexity of a qubit string as the length of the shortest quantum input to a universal quantum Turing machine that produces the initial qubit string with high fidelity. The definition of P. Vitányi (2001, IEEE Trans. Inform. Theory47, 2464–2479) measures the amount of classical information, whereas we consider the amount of quantum information in a qubit string. We argue that our definition is a natural and accurate representation of the amount of quantum information contained in a quantum state. Recently, P. Gács (2001, J. Phys. A: Mathematical and General34, 6859–6880) also proposed two measures of quantum algorithmic entropy which are based on the existence of a universal semidensity matrix. The latter definitions are related to Vitányi's and the one presented in this article, respectively.