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On Poles and Zeros of Linear Quantum Systems

Zhiyuan Dong, Guofeng Zhang, Heung-wing Joseph Lee

TL;DR

The paper develops a comprehensive treatment of zeros and poles in linear quantum systems, establishing a tight link between poles and transmission zeros via $s_0$ and $-s_0^{\ast}$, and tying invariant zeros to system eigenvalues through the Kalman-like structure $\mathcal{A},\mathcal{B},\mathcal{C},\mathcal{D}$. It shows a one-to-one correspondence between poles of the transfer function and transmission zeros, and between eigenvalues of the system matrix $\mathcal{A}$ and invariant zeros, under suitable assumptions. The analysis connects invariant zeros to left invertibility and derives a SISO tradeoff: squeezing benefits from certain zero/pole placements at low frequencies come at the cost of reduced robustness due to sensitivity to plant uncertainty. These results offer a structured, physics-aware lens for control design in quantum technologies, including coherent feedback networks and squeezing applications. Overall, the work clarifies how the algebraic structure imposed by quantum realizability governs zeros, poles, invertibility, and performance in quantum linear systems.

Abstract

The non-commutative nature of quantum mechanics imposes fundamental constraints on system dynamics, which in the linear realm are manifested by the physical realizability conditions on system matrices. These restrictions endow system matrices with special structure. The purpose of this paper is to study such structure by investigating zeros and poses of linear quantum systems. In particular, we show that $-s_0^\ast$ is a transmission zero if and only if $s_0$ is a pole, and which is further generalized to the relationship between system eigenvalues and invariant zeros. Additionally, we study left-invertibility and fundamental tradeoff for linear quantum systems in terms of their zeros and poles.

On Poles and Zeros of Linear Quantum Systems

TL;DR

The paper develops a comprehensive treatment of zeros and poles in linear quantum systems, establishing a tight link between poles and transmission zeros via and , and tying invariant zeros to system eigenvalues through the Kalman-like structure . It shows a one-to-one correspondence between poles of the transfer function and transmission zeros, and between eigenvalues of the system matrix and invariant zeros, under suitable assumptions. The analysis connects invariant zeros to left invertibility and derives a SISO tradeoff: squeezing benefits from certain zero/pole placements at low frequencies come at the cost of reduced robustness due to sensitivity to plant uncertainty. These results offer a structured, physics-aware lens for control design in quantum technologies, including coherent feedback networks and squeezing applications. Overall, the work clarifies how the algebraic structure imposed by quantum realizability governs zeros, poles, invertibility, and performance in quantum linear systems.

Abstract

The non-commutative nature of quantum mechanics imposes fundamental constraints on system dynamics, which in the linear realm are manifested by the physical realizability conditions on system matrices. These restrictions endow system matrices with special structure. The purpose of this paper is to study such structure by investigating zeros and poses of linear quantum systems. In particular, we show that is a transmission zero if and only if is a pole, and which is further generalized to the relationship between system eigenvalues and invariant zeros. Additionally, we study left-invertibility and fundamental tradeoff for linear quantum systems in terms of their zeros and poles.
Paper Structure (8 sections, 6 theorems, 38 equations, 1 figure)

This paper contains 8 sections, 6 theorems, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear quantum systems
  4. Invariant zeros and transmission zeros
  5. Zeros and poles of linear quantum systems
  6. Invariant zeros and left invertibility
  7. A tradeoff in SISO quantum linear systems
  8. Conclusion

Key Result

Lemma II.1

Maciejowski89 Let $G(s)$ be a rational matrix of normal rank $r$. Then $G(s)$ may be transformed by a series of elementary row and column operations into a pseudo-diagonal (can be nonsquare) rational matrix $M(s)$ of the form in which the monic polynomials $\{\alpha_i(s),\beta_i(s)\}$ are coprime for each $i$ and satisfy the divisibility properties In the literature, $M(s)$ is commonly referred

Figures (1)

  • Figure 1: A quantum coherent feedback network composed of two linear quantum systems $G$, $K$, and a beamsplitter.

Theorems & Definitions (23)

  • Remark II.1
  • Remark II.2
  • Lemma II.1
  • Remark II.3
  • Lemma II.2
  • Remark II.4
  • Proposition III.1
  • Example III.1
  • Remark III.1
  • Proposition III.2
  • ...and 13 more