Quantum information distance based on classical descriptions

Classical information distance between two individual finite objects is the length of the shortest program that makes objects convert into each other. This is based on classical Kolmogorov complexity which is an accepted theory of absolute information in bits contained in an individual finite object. Nowadays, quantum Kolmogorov complexity is developed as the absolute information in bits or qubits contained in an individual pure quantum state. In this paper, we give a definition of quantum information distance between two individual pure quantum states based on Vitányi’s definition of quantum Kolmogorov complexity. This extends classical information distance to the quantum domain retaining classical descriptions. We show that our definition is robust, that is, the quantum information distance between two pure quantum states does not depend on the underlying quantum Turing machine.