Measurement back action and a classical uncertainty principle: Heisenberg meets Kalman

Abstract

We study a measurement framework motivated by considering macroscopic (i.e. large, active, and with finite temperature) measurement of microscopic (i.e. small and lossless) but classical dynamics. This unavoidably leads to “measurement back action” on the microscopic dynamics that nevertheless still allows for optimal filtering to minimize estimation error, but with tradeoffs between errors due to estimation and errors due to the back action. We focus on a simple case in which the deterministic effects of the measurement process are completely canceled by active control, and the remaining (coupled) stochastic back action and measurement noise is optimally filtered to minimize estimation error. This leads to a particularly interesting tradeoffs and limits on estimation and back action, analogous in many respects with the Heisenberg uncertainty principle but in an entirely classical framework.