Narrow Results

We have been developing a general theory of information distance and a paradigm of applying this theory to practical problems.[3, 19, 20] There are several problems associated with this theory. On the practical side, among other problems, the strict requirement of triangle inequality is unrealistic in some applications; on the theoretical side, the universality theorems for normalized information distances were only proved in a weak form. In this paper, we will introduce a complementary theory that resolves or avoids these problems.

This article also serves as a brief expository summary for this area. We will tell the stories about how and why some of the concepts were introduced, recent theoretical developments and interesting applications. These applications include whole genome phylogeny, plagiarism detection, document comparison, music classification, language classification, fetal heart rate tracing, question answering, and a wide range of other data mining tasks.

In this paper, we give a definition for quantum information distance. In the classical setting, information distance between two classical strings is developed based on classical Kolmogorov complexity. It is defined as the length of a shortest transition program between these two strings in a universal Turing machine. We define the quantum information distance based on Berthiaume et al.’s quantum Kolmogorov complexity. The quantum information distance between qubit strings is defined as the length of the shortest quantum transition program between these two qubit strings in a universal quantum Turing machine. We show that our definition of quantum information distance is invariant under the choice of the underlying quantum Turing machine.