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Instantaneous Sobolev Regularization for Dissipative Bosonic Dynamics

Pablo Costa Rico, Paul Gondolf, Tim Möbus

TL;DR

<3-5 sentence high-level summary>

Abstract

We investigate quantum Markov semigroups on bosonic Fock space and identify a broad class of infinite-dimensional dissipative evolutions that exhibit instantaneous Sobolev-regularization. Motivated by stability problems in quantum computation, we show that for certain Lindblad operators that are polynomials of creation and annihilation operators, the resulting dynamics immediately transform any initial state into one with finite expectation in all powers of the number operator. A key application is in the bosonic cat code, where we obtain explicit estimates in the trace norm for the speed of convergence. These estimates sharpen existing perturbative bounds at both short and long times, offering new analytic tools for assessing stability and error suppression in bosonic quantum information processing. For example, we improve the strong exponential convergence of the (shifted) $2$-photon dissipation to its fixed point to the uniform topology.

Instantaneous Sobolev Regularization for Dissipative Bosonic Dynamics

TL;DR

<3-5 sentence high-level summary>

Abstract

We investigate quantum Markov semigroups on bosonic Fock space and identify a broad class of infinite-dimensional dissipative evolutions that exhibit instantaneous Sobolev-regularization. Motivated by stability problems in quantum computation, we show that for certain Lindblad operators that are polynomials of creation and annihilation operators, the resulting dynamics immediately transform any initial state into one with finite expectation in all powers of the number operator. A key application is in the bosonic cat code, where we obtain explicit estimates in the trace norm for the speed of convergence. These estimates sharpen existing perturbative bounds at both short and long times, offering new analytic tools for assessing stability and error suppression in bosonic quantum information processing. For example, we improve the strong exponential convergence of the (shifted) -photon dissipation to its fixed point to the uniform topology.

Paper Structure

This paper contains 18 sections, 29 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sobolev-regularizing QMS
  4. Single-mode regularizing QMS
  5. Mulit-mode: concentrated moment propagation bounds
  6. Intermediate and large time perturbation bounds
  7. Single-mode applications
  8. Families of photon dissipation
  9. Exponential convergence in $1\rightarrow1$ norm
  10. Perturbation theory in operator norm
  11. Multi-mode applications
  12. Multi-mode photon dissipation
  13. Local perturbation perturb locally
  14. Conclusion
  15. ODE comparison lemma
  16. ...and 3 more sections

Key Result

Lemma 3.1

Let $y : [0, \infty) \to [0, \infty)$ be a continuously differentiable function satisfying the differential inequality for constants $a, b > 0$ and exponent $p > 1$. Then,

Theorems & Definitions (60)

  • Definition 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2: Jensen’s inequality for moments
  • proof
  • Theorem 3.3
  • proof
  • Remark 1
  • Remark 2
  • Lemma 3.5: Jensen’s inequality for multi-mode moments
  • ...and 50 more