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Wiener Filtering for Passive Linear Quantum Systems

V. Ugrinovskii, M. R. James

TL;DR

This paper extends Wiener filtering to passive quantum linear systems to address coherent equalization of quantum channels, revealing that physical realizability imposes nonconvex constraints and introduces a threshold on input noise variance for mean-square error improvement. It develops two relaxation-based approaches to design quantum Wiener filters and illustrates them with two setups: a static optical beam splitter and a dynamical optical cavity. The beam-splitter analysis shows a clear noise-threshold below which no MSE improvement is possible and, above it, explicit constant-valued filters with added noise achieve reduction in error. The Wiener-Hopf approach for the cavity yields a dynamical optimal $H_{11}(s)$ and demonstrates how, under sufficient noise variance, a fully physically realizable filter can be constructed with explicit transfer-function components. Overall, the results highlight fundamental differences from classical Wiener filtering and provide practical design guidance for coherent quantum communication systems.

Abstract

This paper considers a version of the Wiener filtering problem for equalization of passive quantum linear quantum systems. We demonstrate that taking into consideration the quantum nature of the signals involved leads to features typically not encountered in classical equalization problems. Most significantly, finding a mean-square optimal quantum equalizing filter amounts to solving a nonconvex constrained optimization problem. We discuss two approaches to solving this problem, both involving a relaxation of the constraint. In both cases, unlike classical equalization, there is a threshold on the variance of the noise below which an improvement of the mean-square error cannot be guaranteed.

Wiener Filtering for Passive Linear Quantum Systems

TL;DR

This paper extends Wiener filtering to passive quantum linear systems to address coherent equalization of quantum channels, revealing that physical realizability imposes nonconvex constraints and introduces a threshold on input noise variance for mean-square error improvement. It develops two relaxation-based approaches to design quantum Wiener filters and illustrates them with two setups: a static optical beam splitter and a dynamical optical cavity. The beam-splitter analysis shows a clear noise-threshold below which no MSE improvement is possible and, above it, explicit constant-valued filters with added noise achieve reduction in error. The Wiener-Hopf approach for the cavity yields a dynamical optimal and demonstrates how, under sufficient noise variance, a fully physically realizable filter can be constructed with explicit transfer-function components. Overall, the results highlight fundamental differences from classical Wiener filtering and provide practical design guidance for coherent quantum communication systems.

Abstract

This paper considers a version of the Wiener filtering problem for equalization of passive quantum linear quantum systems. We demonstrate that taking into consideration the quantum nature of the signals involved leads to features typically not encountered in classical equalization problems. Most significantly, finding a mean-square optimal quantum equalizing filter amounts to solving a nonconvex constrained optimization problem. We discuss two approaches to solving this problem, both involving a relaxation of the constraint. In both cases, unlike classical equalization, there is a threshold on the variance of the noise below which an improvement of the mean-square error cannot be guaranteed.

Paper Structure

This paper contains 8 sections, 3 theorems, 46 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Open linear passive quantum systems
  3. Equalization problem for annihilation-only communication systems
  4. Constraint relaxation
  5. Two approaches to quantum Wiener equalization
  6. Equalization via optimization of power spectrum density: An optical beam splitter
  7. The Wiener-Hopf technique for quantum equalization: An equalizer for an optical cavity
  8. Conclusions

Key Result

Proposition 1

The corresponding expressions for the optimal error power spectrum density are

Figures (3)

  • Figure 1: A general quantum communication system. The transfer function $G(s)$ represents the channel, and $H(s)$ represents an equalizing filter.
  • Figure 2: A beam splitter and a quantum equalizer system.
  • Figure 3: A cavity, beam splitters and an equalizer system.

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 2
  • Proposition 3