Lectures on Quantum Measurement Theory: III

Lecture III: Measurement error, disturbance, the universally valid reformulation of Heisenberg’s uncertainty principle, and a quantitative generalization of the Wigner–Araki–Yanase theorem

Definitions of measurement error and disturbance are introduced (Ozawa 2002, 2019) and it is shown that there exists a solvable model for a physically realizable measurement that serves as a counterexample both to Heisenberg’s uncertainty principle in the conventional formulation and to the SQL (Ozawa 1988, 1989, 2002). Thus, those limits are no more considered as universal limits. In fact, the above counter example to SQL was found in 1988 using the idea of contractive state measurements by Yuen (1983) and the LIGO was started in 1994 to succeed in the gravitational wave detection in 2015 as announced in 2016.

New formulations are then proved for the uncertainty principle concerning the errors in the approximate simultaneous measurement of two physical quantities, called the "joint error relation" (Ozawa 2003b, 2004), and for the uncertainty principle concerning the error and disturbance associated with the measurement of a single physical quantity, called the "error-disturbance relation" (Ozawa 2003a). From the error-disturbance relation, a quantitative relation for measurement error under an additive conservation law is proved (Ozawa 2002a, 2003b), generalizing the "Wigner–Araki–Yanase theorem" (Wigner 1952, Araki-Yanase 1960), which states that a physical quantity not commuting with a conserved quantity cannot be measured accurately by a measurement interaction satisfying an additive conservation law. The above relation also derives limits for realizing quantum computing and operations under conservation laws (Ozawa 2002b), the results later developed as the resource theory of asymmetry.