  This paper is concerned with the coherent quantum linear-quadratic-Gaussian
control problem of minimising an infinite-horizon mean square cost for a
measurement-free field-mediated interconnection of a quantum plant with a
stabilising quantum controller. The plant and the controller are multimode open
quantum harmonic oscillators, governed by linear quantum stochastic
differential equations and coupled to each other and the external multichannel
bosonic fields in the vacuum state. We discuss an interplay between the quantum
physical realizability conditions and the Luenberger structure associated with
the classical separation principle. This leads to a quadratic constraint on the
controller gain matrices, which is formulated in the framework of a swapping
transformation for the conjugate positions and momenta in the canonical
representation of the controller variables. For the class of coherent quantum
controllers with the Luenberger dynamics, we obtain first-order necessary
conditions of optimality in the form of algebraic equations, involving a
matrix-valued Lagrange multiplier.
