  The purpose of this tutorial is to give a brief introduction to linear
quantum control systems. The mathematical model of linear quantum control
systems is presented first, then some fundamental control-theoretic notions
such as stability, controllability and observability are given, which are
closely related to several important concepts in quantum information science
such as decoherence-free subsystems, quantum non-demolition variables, and
back-action evasion measurements. After that, quantum Gaussian states are
introduced, in particular, an information-theoretic uncertainty relation is
presented which often gives a better bound for mixed Gaussian states than the
well-known Heisenberg uncertainty relation. The quantum Kalman filter is
presented for quantum linear systems, which is the quantum analogy of the
Kalman filter for classical (namely, non-quantum-mechanical) linear systems.
The quantum Kalman canonical decomposition for quantum linear systems is
recorded, and its application is illustrated by means of a recent experiment.
As single- and multi-photon states are useful resources in quantum information
technology, the response of quantum linear systems to these types of input is
presented. Finally, coherent feedback control of quantum linear systems is
briefly introduced, and a recent experiment is used to demonstrate the
effectiveness of quantum linear systems and networks theory.dback control of
quantum linear systems is briefly introduced, and a recent experiment is used
to demonstrate the effectiveness of quantum linear systems and networks theory.
