Quantum Zeno effect
= Quantum Zeno effect
Inhibition of a transition by increasingly frequent ideal <projective measurements> that check whether a <quantum system> remains in its initial <quantum state>. For a normalized vector $\psi$, a fixed <Hamiltonian operator> $H$ and the <orthogonal projection> $P=|\psi\rangle\langle\psi|$, the <probability> of passing every check at $N$ equally spaced times up to fixed time $T$ is
$$
\mathbb P_N(T)=\left|\langle\psi,e^{-iHT/(N\hbar)}\psi\rangle\right|^{2N}.
$$
For a finite-dimensional <Hilbert space>, the <quadratic short-time quantum survival probability> implies $\mathbb P_N(T)\to1$. This is an ideal limit with instantaneous measurements and <unitary time evolution> between them.