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.
