Informatic error-disturbance relation in the qubit case

Abstract

In 1927, Heisenberg heuristically disclosed the tradeoff between the error in the measurement and the caused disturbance on another complementary observable. In the quantum theory, most of uncertainty relations are proposed to reveal the amount of unavoidable uncertainty in the measuring process. In this paper, we study the error-disturbance relation from the information viewpoint. We ask how much information, rather than how much uncertainty, can be obtained during the two sequential measurements. To achieve optimal information gain, we argue that the strategy for the "intelligent" prior apparatus is to clone the unknown state, and for the posterior one is to perform the swapping operation. We propose the coarse-grained random access code, and therein information causality as a physical principle can be exploited for deriving the upper-bound of information gain. Finally, we conjecture the information gain of measuring the position and momentum of a quantum object in the coarse-grained way.