Feedback control engineers have been interested in MIMO (multiple-input multiple-output) extensions of SISO (single-input single-output) results of various kinds due to its rich mathematical structure and practical applications. An outstanding proble
m in quantum feedback control is the extension of the SISO theory of Markovian feedback by Wiseman and Milburn [Phys. Rev. Lett. {bf 70}, 548 (1993)] to multiple inputs and multiple outputs. Here we generalize the SISO homodyne-mediated feedback theory to allow for multiple inputs, multiple outputs, and emph{arbitrary} diffusive quantum measurements. We thus obtain a MIMO framework which resembles the SISO theory and whose additional mathematical structure is highlighted by the extensive use of vector-operator algebra.
The master equation for the state of an open quantum system can be unravelled into stochastic trajectories described by a stochastic master equation. Such stochastic differential equations can be interpreted as an update formula for the system state
conditioned on results obtained from monitoring the bath. So far only one parameterization (mathematical representation) for arbitrary diffusive unravellings (quantum trajectories arising from monitorings with Gaussian white noise) of a system described by a master equation with $L$ Lindblad terms has been found [H. M. Wiseman and A. C. Doherty, Phys. Rev. Lett. {bf 94}, 070405 (2005)]. This parameterization, which we call the U-rep, parameterizes diffusive unravellings by $L^2+2L$ real numbers, arranged in a matrix ${sf U}$ subject to three constraints. In this paper we investigate alternative parameterizations of diffusive measurements. We find, rather surprisingly, the description of diffusive unravellings can be unified by a single equation for a non-square complex matrix if one is willing to allow for some redundancy by lifting the number of real parameters necessary from $L^2+2L$ to $3L^2+L$. We call this parameterization the M-rep. Both the M-rep and U-rep lack a physical picture of what the measurement should look like. We thus propose another parameterization, the B-rep, that details how the measurement is implemented in terms of beam-splitters, phase shifters, and homodyne detectors. Relations between the different representations are derived.
