Coherent Equalization of Linear Quantum Systems

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.