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Design of Coherent Passive Quantum Equalizers Using Robust Control Theory

V. Ugrinovskii, M. R. James

Abstract

The paper develops a methodology for the design of coherent equalizing filters for quantum communication channels. Given a linear quantum system model of a quantum communication channel, the aim is to obtain another quantum system which, when coupled with the original system, mitigates degrading effects of the environment. The main result of the paper is a systematic equalizer synthesis algorithm which relies on methods of state-space robust control design via semidefinite programming.

Design of Coherent Passive Quantum Equalizers Using Robust Control Theory

Abstract

The paper develops a methodology for the design of coherent equalizing filters for quantum communication channels. Given a linear quantum system model of a quantum communication channel, the aim is to obtain another quantum system which, when coupled with the original system, mitigates degrading effects of the environment. The main result of the paper is a systematic equalizer synthesis algorithm which relies on methods of state-space robust control design via semidefinite programming.
Paper Structure (11 sections, 8 theorems, 87 equations, 8 figures)

This paper contains 11 sections, 8 theorems, 87 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Coherent equalization problem for quantum linear systems
  3. An open quantum system model of a quantum communication channel
  4. Coherent equalization problem
  5. A convex relaxation and a physically realizable upper bound
  6. Suboptimal solutions to the auxiliary problems
  7. Synthesis of suboptimal physically realizable equalizers
  8. Examples
  9. Equalization of an optical cavity system
  10. Equalization of a two-input two-output four-noise system comprised of two cavities
  11. Conclusions

Key Result

Theorem 1

Suppose $\gamma>0$ is such that $\mathcal{H}_{11,\gamma}^-$ is not empty. Then every element $H_{11}(s)$ of $\mathcal{H}_{11,\gamma}^-$ satisfies all three conditions (H1)-(H3). As a result, an admissible physically realizable filter transfer function $H(s)\in\mathcal{H}_p$ can be constructed using

Figures (8)

  • Figure 1: A general quantum communication system.
  • Figure 2: A cavity, beam splitters and an equalizer system.
  • Figure 3: The magnitude Bode plot of the suboptimal transfer function $H_{11}(s)$ in equation (\ref{['eq:56']}).
  • Figure 4: Power spectrum densities $P_{y-u}$ and $P_{e}$ and the suboptimal value $\gamma^2$. The circles indicate the values $P_e(i\omega_l)$ obtained from the solutions of the optimization problem (\ref{['eq:71.LMI']}).
  • Figure 5: A cavity and beam splitters realization of the equalizer.
  • ...and 3 more figures

Theorems & Definitions (16)

  • Definition 1
  • Remark 1
  • Definition 2: UJ2aUJ2
  • Remark 2
  • Theorem 1
  • Theorem 2
  • Remark 3
  • Lemma 1
  • Remark 4
  • Lemma 2
  • ...and 6 more