Abstract: The Kalman canonical form is a standard result in classical linear systems theory. In this talk, we discuss quantum linear systems from a control-theoretic point of view. After that we present the quantum Kalman canonical form. As an application, we show how it can be used to study decoherence-free modes, quantum non-demolition variables, and back-action evading measurements in quantum information science. The proposed theory will be demonstrated by several experiments from quantum optics and opto-mechanical systems, including:
1. Optomechanical dark mode, Dong et al., Science, 338:609--1613, 2012.
2. Quantum backaction evading measurement of collective mechanical modes, Ockeloen-Korppi, et al., Physical Review Letters, 117:140401, 2016.
3. Light-mediated strong coupling between a mechanical oscillator and atomic spins 1 meter apart, Karg, et al., Science 369:174–179, 2020.
4. Quantum mechanics–free subsystem with mechanical oscillators, de Lepinay, et al., Science 372:625–629, 2021.