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, for example, 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 mathcal {O}(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.