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A two-disk approach to the synthesis of coherent passive equalizers for linear quantum systems

Valery Ugrinovskii, Shuixin Xiao

TL;DR

The paper tackles the problem of designing coherent, physically realizable, passive equalizers for linear quantum channels to minimize mean-square error. It recasts the auxiliary two-disk $H_{\infty}$ problem via spectral factorization of the input-perturbed channel transfer, and employs Youla parameterization to convert controller design into a convex LMI feasibility problem. An explicit Algorithm 1 is provided to construct a transfer function $H(s)$ that guarantees $\bar{\boldsymbol{\sigma}}(P_e(i\omega))<\gamma^2$ for all frequencies, with a practical pathway to derive all components of the equalizer transfer. Compared with prior work, the method relaxes restrictive conditions and expands applicability, delivering near-optimal performance in numerical examples, including low-noise regimes where earlier approaches fail, while remaining implementable in quantum optical hardware."

Abstract

The coherent equalization problem consists in designing a quantum system acting as a mean-square near optimal filter for a given quantum communication channel. The paper develops an improved method for the synthesis of transfer functions for such equalizing filters, based on a linear quantum system model of the channel. The method draws on a connection with the two-disk problem of ${H}_{\infty}$ control for classical (i.e., nonquantum) linear uncertain systems. Compared with the previous methods, the proposed method applies to a broader class of linear quantum communication channels.

A two-disk approach to the synthesis of coherent passive equalizers for linear quantum systems

TL;DR

The paper tackles the problem of designing coherent, physically realizable, passive equalizers for linear quantum channels to minimize mean-square error. It recasts the auxiliary two-disk problem via spectral factorization of the input-perturbed channel transfer, and employs Youla parameterization to convert controller design into a convex LMI feasibility problem. An explicit Algorithm 1 is provided to construct a transfer function that guarantees for all frequencies, with a practical pathway to derive all components of the equalizer transfer. Compared with prior work, the method relaxes restrictive conditions and expands applicability, delivering near-optimal performance in numerical examples, including low-noise regimes where earlier approaches fail, while remaining implementable in quantum optical hardware."

Abstract

The coherent equalization problem consists in designing a quantum system acting as a mean-square near optimal filter for a given quantum communication channel. The paper develops an improved method for the synthesis of transfer functions for such equalizing filters, based on a linear quantum system model of the channel. The method draws on a connection with the two-disk problem of control for classical (i.e., nonquantum) linear uncertain systems. Compared with the previous methods, the proposed method applies to a broader class of linear quantum communication channels.

Paper Structure

This paper contains 11 sections, 8 theorems, 104 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Open quantum linear systems
  4. Quantum coherent equalization problem
  5. A two-disk formulation of the auxiliary problem
  6. An LMI synthesis of two-disk $H_{\infty}$ controllers
  7. Synthesis of coherent equalizers
  8. A solution to the auxiliary two-disk problem \ref{['eq31']}
  9. An algorithm for synthesis of a coherent equalizer
  10. Illustrative example
  11. Conclusions

Key Result

Lemma 1

A stable proper transfer function $H_{11}(s)$ which is analytic in a half-plane $\operatorname{Re}s >-\tau\;(\exists \tau>0)$ satisfies eq12 if and only if where

Figures (8)

  • Figure 1: A general quantum equalization system; e.g., see Ugrinovskii2024.
  • Figure 2: The $H_\infty$ control setting for the auxiliary problem.
  • Figure 3: A cavity, beam splitters and an equalizer system from Ugrinovskii2024a.
  • Figure 4: The Bode plots of the transfer function $H_{11}(s)$ in equation \ref{['eq83']} (the solid black line) and the corresponding transfer function in Ugrinovskii2024a (the dashed magenta line) for $\sigma_{w_2}^2=3$.
  • Figure 5: The power spectrum density $P_e(i\omega)$ for the system with the equalizer \ref{['eq83']} (the solid blue line). Also shown in the figure are the power spectrum density $P_{y-u}(i\omega)$ of the difference $y-u$ (the dashed magenta line), the power spectrum density $P_e(i\omega)$ for the system with the equalizer obtained in Ugrinovskii2024a (the dash-dotted red line) for this $\gamma^2$, and the values obtained from the semidefinite program \ref{['eq74']} (the circles).
  • ...and 3 more figures

Theorems & Definitions (10)

  • Definition 1: Ugrinovskii2024a, Definition 1
  • Definition 2
  • Lemma 1: Lemma 2, Ugrinovskii2024a
  • Theorem 1
  • Lemma 2
  • Theorem 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 3