[Paper Review] Energy preserving evolutions over Bosonic systems

Paul Gondolf, Tim Möbus+1 more|arXiv (Cornell University)|Jul 25, 2023

Quantum Computing Algorithms and ArchitectureComputer Science3 citations

TL;DR

This paper establishes a generation theorem for quantum Markov semigroups on continuous-variable systems that preserve Sobolev-type moment bounds, enabling rigorous perturbation analysis of energy-constrained dynamics. It proves that generators in GKSL form with polynomial jump operators and controlled moment growth generate trace-preserving semigroups, and derives quantitative continuity bounds for energy-constrained capacities and steady states of quantum Ornstein-Uhlenbeck and CAT qubit systems.

ABSTRACT

The exponential convergence to invariant subspaces of quantum Markov semigroups plays a crucial role in quantum information theory. One such example is in bosonic error correction schemes, where dissipation is used to drive states back to the code-space - an invariant subspace protected against certain types of errors. In this paper, we investigate perturbations of quantum dynamical semigroups that operate on continuous variable (CV) systems and admit an invariant subspace. First, we prove a generation theorem for quantum Markov semigroups on CV systems under the physical assumptions that (i) the generator is in GKSL form with corresponding jump operators defined as polynomials of annihilation and creation operators; and (ii) the (possibly unbounded) generator increases all moments in a controlled manner. Additionally, we show that the level sets of operators with bounded first moments are admissible subspaces of the evolution, providing the foundations for a perturbative analysis. Our results also extend to time-dependent semigroups and multi-mode systems. We apply our general framework to two settings of interest in continuous variable quantum information processing. First, we provide a new scheme for deriving continuity bounds on the energy-constrained capacities of Markovian perturbations of quantum dynamical semigroups. Second, we provide quantitative perturbation bounds for the steady state of the quantum Ornstein-Uhlenbeck semigroup and the invariant subspace of the photon dissipation used in bosonic error correction.

Motivation & Objective

• To establish a rigorous mathematical framework for quantum Markov semigroups on continuous-variable systems that preserve moment bounds and invariant subspaces.
• To address the failure of standard GKSL form in unbounded settings by imposing moment control conditions on the generator.
• To enable perturbation analysis of energy-constrained quantum channels and steady states in bosonic systems.
• To derive quantitative bounds on the deviation of energy-constrained capacities and steady states under Lindbladian perturbations.
• To extend the theory to time-dependent evolutions and multi-mode systems, supporting applications in quantum error correction and quantum information processing.

Proposed method

• Prove a generation theorem for strongly continuous quantum Markov semigroups on bosonic systems using GKSL-form generators with polynomial jump operators in annihilation/creation operators.
• Introduce a moment control condition ensuring that the generator increases all moments in a controlled way, enabling Sobolev space preservation.
• Define admissible subspaces as level sets of operators with bounded first moments, forming invariant sets under the evolution.
• Establish equivalence between energy-constrained diamond norms and weighted Sobolev norms via Lemma 6.5, linking channel distance to Sobolev operator norms.
• Apply perturbation theory to Gaussian and non-Gaussian semigroups, deriving bounds on the diamond norm distance of perturbed channels.
• Use the framework to derive continuity bounds for energy-constrained capacities and steady-state convergence in quantum Ornstein-Uhlenbeck and photon-dissipation dynamics.

Experimental results

Research questions

• RQ1 Under what conditions does a GKSL-form generator with unbounded jump operators generate a trace-preserving quantum Markov semigroup on continuous-variable systems?
• RQ2 How can moment growth of the generator be controlled to ensure preservation of Sobolev-type moment bounds?
• RQ3 What are the quantitative continuity bounds for energy-constrained quantum channel capacities under Lindbladian perturbations?
• RQ4 How do perturbations affect the steady state of the quantum Ornstein-Uhlenbeck semigroup and the CAT qubit code space?
• RQ5 Can the framework be extended to time-dependent semigroups and multi-mode bosonic systems with invariant subspaces?

Key findings

• A new generation theorem is proven for quantum Markov semigroups on continuous-variable systems, showing that GKSL generators with polynomial jump operators and controlled moment growth generate trace-preserving evolutions.
• The level sets of operators with bounded first moments form admissible invariant subspaces, enabling perturbative analysis of dynamics restricted to these subspaces.
• For Gaussian perturbations of the quantum Ornstein-Uhlenbeck semigroup, the diamond norm distance between the original and perturbed channels is bounded by $ (1+E) u imes C( u) imes ext{max}"](1, D( u)) $, where $ u $ is the perturbation strength.
• The energy-constrained diamond norm distance between two channels is shown to be equivalent to a weighted Sobolev norm of the difference channel, as formalized in Lemma 6.5.
• Perturbation bounds are derived for the steady state of the photon-dissipation semigroup used in CAT qubit error correction, ensuring convergence under small perturbations.
• The framework enables continuity bounds on energy-constrained capacities of Markovian perturbations, with applications to quantum communication and sensing.