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Optimal convergence rate in the quantum Zeno effect for open quantum systems in infinite dimensions

Tim Möbus, Cambyse Rouzé

TL;DR

This work establishes the optimal $O\left(\tfrac{1}{n}\right)$ convergence rate for quantum Zeno dynamics in open, infinite-dimensional systems by developing a modified Chernoff bound and handling unbounded generators. It proves two main results: a spectral-gap-type bound under uniform power convergence and a uniform-topology result for finitely many unit-circle eigenvalues, both with explicit error terms and broad applicability. The authors provide explicit bounds, show optimality via concrete examples, and demonstrate the framework on finite- and infinite-dimensional models, including harmonic oscillators and bosonic beam-splitter channels. The findings advance understanding of Zeno dynamics in realistic open systems and offer tools for decoherence suppression and quantum control with rigorous rates.

Abstract

In open quantum systems, the quantum Zeno effect consists in frequent applications of a given quantum operation, e.g.~a measurement, used to restrict the time evolution (due e.g.~to decoherence) to states that are invariant under the quantum operation. In an abstract setting, the Zeno sequence is an alternating concatenation of a contraction operator (quantum operation) and a $C_0$-contraction semigroup (time evolution) on a Banach space. In this paper, we prove the optimal convergence rate of order $\tfrac{1}{n}$ of the Zeno sequence by proving explicit error bounds. For that, we derive a new Chernoff-type $\sqrt{n}$-Lemma, which we believe to be of independent interest. Moreover, we generalize the convergence result for the Zeno effect in two directions: We weaken the assumptions on the generator, inducing the Zeno dynamics generated by an unbounded generator and we improve the convergence to the uniform topology. Finally, we provide a large class of examples arising from our assumptions.

Optimal convergence rate in the quantum Zeno effect for open quantum systems in infinite dimensions

TL;DR

This work establishes the optimal convergence rate for quantum Zeno dynamics in open, infinite-dimensional systems by developing a modified Chernoff bound and handling unbounded generators. It proves two main results: a spectral-gap-type bound under uniform power convergence and a uniform-topology result for finitely many unit-circle eigenvalues, both with explicit error terms and broad applicability. The authors provide explicit bounds, show optimality via concrete examples, and demonstrate the framework on finite- and infinite-dimensional models, including harmonic oscillators and bosonic beam-splitter channels. The findings advance understanding of Zeno dynamics in realistic open systems and offer tools for decoherence suppression and quantum control with rigorous rates.

Abstract

In open quantum systems, the quantum Zeno effect consists in frequent applications of a given quantum operation, e.g.~a measurement, used to restrict the time evolution (due e.g.~to decoherence) to states that are invariant under the quantum operation. In an abstract setting, the Zeno sequence is an alternating concatenation of a contraction operator (quantum operation) and a -contraction semigroup (time evolution) on a Banach space. In this paper, we prove the optimal convergence rate of order of the Zeno sequence by proving explicit error bounds. For that, we derive a new Chernoff-type -Lemma, which we believe to be of independent interest. Moreover, we generalize the convergence result for the Zeno effect in two directions: We weaken the assumptions on the generator, inducing the Zeno dynamics generated by an unbounded generator and we improve the convergence to the uniform topology. Finally, we provide a large class of examples arising from our assumptions.
Paper Structure (16 sections, 29 theorems, 169 equations, 3 figures)

This paper contains 16 sections, 29 theorems, 169 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Chernoff Lemma and Trotter-Kato's Product Formula
  5. Strongly Continuous Zeno Dynamics
  6. Proof of Theorem 3.1
  7. Upper bound on \ref{['eq:proofthm1-term1']}:
  8. Proof of Proposition 3.1 and Corollary 3.2
  9. Proof of Proposition 5.7
  10. Uniform Power Convergence with Finitely Many Eigenvalues
  11. Proof of Theorem 3.3
  12. Examples
  13. Discussion and Open Questions
  14. Holomorphic Functional Calculus and Semicontinuous Spectra
  15. Spectral Gap Assumption
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $(\mathcal{L},\mathcal{D}(\mathcal{L}))$ be the generator of the $C_0$-semigroup $(e^{t\mathcal{L}})_{t\geq0}$ defined on $\mathcal{X}$. Then the following properties hold:

Figures (3)

  • Figure 1:
  • Figure 2:
  • Figure 3:

Theorems & Definitions (66)

  • Lemma 2.1: [8, Lem. II.1.3]
  • Lemma 2.2: Integral equation for semigroups
  • proof
  • Corollary 2.3
  • proof
  • Theorem I: stated as \ref{['thm:spectral-gap']} in main text
  • Remark
  • Example 1
  • Proposition 3.1
  • Corollary 3.2
  • ...and 56 more