Lectures on Quantum Measurement Theory: I

Lecture I: Conventional approach: Repeatability, Heisenberg’s original uncertainty principle, and the SQL for gravitational-wave detection

The conventional approach to quantum measurement theory taken by von Neumann (1932), Dirac (1958), and Schrödinger (1935) assumes the "repeatability hypothesis" stating that if a physical quantity is measured twice in succession, then the same value is obtained each time, which is often quantitatively generalized to the "approximately repeatable hypothesis" stating that after a measurement of a physical quantity with error ε, the post-measurement deviation around the measured value is no larger than ε; this is equivalent to saying that the state after obtaining a measurement result with error ε becomes an ε-approximate eigenstate corresponding to that measurement result.

From the approximate repeatability hypothesis, one can derive "Heisenberg’s original formulation of the uncertainty principle," namely, that when position and momentum are approximately measured simultaneously, the product of their respective errors is at least ℏ/2 (Heisenberg 1927, Kennard 1927, Ozawa 2015), as well as the "standard quantum limit (SQL) for monitoring the free-mass position", which states that when the position of a free mass m is measured at a time interval τ, the result of the second measurement cannot be predicted with uncertainty smaller than (ℏτ/ m)^{1/2} (Caves 1985). The last result leads to a sensitivity limit for interferometric gravitational-wave detectors, and in the early 1980s it was therefore argued that gravitational waves of the expected strength could not be observed using interferometric detectors (Braginsky et al. 1980, Caves et al. 1980).