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

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 $\frac{1}{n}$ 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.

27 pages

References in corpus (5)

• Quantum Zeno dynamics: mathematical and physical aspects
• Complete entropic inequalities for quantum Markov chains
• Quantum evolution in the stroboscopic limit of repeated measurements
• Operator-Norm Convergence of the Trotter Product Formula on Hilbert and Banach Spaces: A Short Survey
• Mathematics of the Quantum Zeno Effect

Cited by in corpus (7)

• Unification of Random Dynamical Decoupling and the Quantum Zeno Effect
• On Strong Bounds for Trotter and Zeno Product Formulas with Bosonic Applications
• Ultrastrong coupling, nonselective measurement and quantum Zeno dynamics
• Convergence Rates for the Trotter Splitting for Unbounded Operators
• Energy preserving evolutions over Bosonic systems
• Bath Dynamical Decoupling with a Quantum Channel
• Learning and simulating bosonic systems via finite-energy locality