Information-Theoretical Measure of Distance by Complementary Measurement

Abstract

Mathematical formulations of all the fundamental physical theories are based on the concept of an abstract space. The structure of the space and the theories is characterized in terms of its metric. For example, the Minkowski metric defines the mathematicalstructure of the special theory of relativity and the Rieman metric defines the structure of thegeneral theory of relativity. What is a natural measure of distance for quantum theory? In particular, two metrics have been applied to the wide real m of quantum information processing: Hilbert-Schmidt distance and Bures metric which corresponds to fidelity. Here, we propose an OPERATIONAL measure of distance of two quantum states, which conversely tells us their closeness. This is defined as a sum of differences in partial knowledge over a complete set of mutually complementary measurements for the two states. It is shown that the measure is operationally invariant and it is equivalent to the Hilbert-Schmidt distance. The operational measure of distance provides a remarkable interpretation of the information distance between quantum states.