Coherent Equalization of Linear Quantum Systems

TL;DR

An additional semidefinite relaxation combined with the Nevanlinna-Pick interpolation is shown to lead to a tractable algorithm for the design of a suboptimal coherent equalizer.

Abstract

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.

Figures (10)

• Figure 1: Example quantum optical communication system consisting of a channel and equalizer, modelled by optical beamsplitters and signals. The message signal $\breve{u}$ is passed through the channel which may degrade the message, resulting in the received signal $\breve{y}$. The equalizer near-optimally recovers the message in the mean square sense, producing an improved signal $\breve{\hat{u}}$.
• Figure 2: A general quantum communication system. The transfer function $\Gamma(s)$ represents the channel, and $\Xi(s)$ represents an equalizing filter.
• Figure 3: Power spectrum densities $P_{y-u}$ and $P_{e}$ (given by (\ref{['eq:7']})) and the optimal value of the SDP problem (\ref{['eq:71.LMI']})--(\ref{['eq:14']}) for a range of $\sigma_w^2$, for the beamsplitter transmittance of $\eta=0.7$.
• Figure 4: The theoretical (equation (\ref{['eq:5']})) and numerically obtained (via the SDP problem (\ref{['eq:71.LMI']})--(\ref{['eq:14']})) optimal gains $|H_{11}|$ for a range of $\sigma_w^2$, for the beamsplitter transmittance of $\eta=0.7$.
• Figure 5: A cavity, beamsplitters and an equalizer system.

Theorems & Definitions (18)

• Lemma 1
• Lemma 2
• Lemma 3: Youla, Theorem 2 of Youla-1961
• Theorem 1
• Remark 1
• Remark 2
• Theorem 2
• Theorem 3
• Corollary 1
• Corollary 2