  This paper introduces a $H_\infty$-like methodology of coherent filtering for
equalization of passive linear quantum systems to help mitigate degrading
effects of quantum communication channels. For such systems, which include a
wide range of linear quantum optical devices and signals, we seek to find a
near optimal equalizing filter which is itself a passive quantum system. The
problem amounts to solving an optimization problem subject to constraints
dictated by the requirement for the equalizer to be physically realizable. By
formulating these constraints in the frequency domain, we show that the problem
admits a convex $H_\infty$-like formulation. This allows us to derive a set of
suboptimal coherent equalizers using $J$-spectral factorization. An additional
semidefinite relaxation combined with the Nevanlinna-Pick interpolation is
shown to lead to a tractable algorithm for the design of a near optimal
coherent equalizer.
