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 $P=|\psi ⟩⟨\psi |$, the probability of passing every check at $N$ equally spaced times up to fixed time $T$ isFor a finite-dimensional Hilbert space, the quadratic short-time quantum survival probability implies ${\mathbb{P}}_N(T)\to 1$. This is an ideal limit with instantaneous measurements and unitary time evolution between them.

In 1990, Wayne M. Itano, Daniel J. Heinzen, J. J. Bollinger and David J. Wineland reported inhibited transitions in about 5,000 ${}^9{\mathrm{Be}}^{+}$ ions held in a cylindrical Penning trap. Trapped magnesium ions, illuminated at 280 nm, kept the beryllium ions cold. A 320.7 MHz radio frequency field coupled two ground-state hyperfine structure sublevels during a 256 ms pulse adjusted for complete transfer. Up to 64 intervening 313 nm pulses addressed an optical transition from level 1: scattering distinguished it from level 2. Afterward, calibrated fluorescence counts over 100 ms measured the final quantum state population. This ensemble readout did not select ions that remained in the initial quantum state at every check; departure followed by return also contributes. The apparatus made inhibition by repeated quantum measurements experimentally accessible.^[ref]

Start in $a$, drive resonantly with Hamiltonian operator $\hslash \Omega X/2$, and perform $n$ ideal basis measurements separated by $T/n$. An interval changes either basis state with probability $p={\sin }^2(\Omega T/(2n))$. Ignoring measurement records leaves a two-state Markov chain whose population difference is multiplied by $1-2p=\cos (\Omega T/n)$ each step. This proves the displayed final-population formula.

For a pi pulse, $\Omega T=\pi$, it is ${\pi }^2/(4n)+O(n^{-2})$. The repeated projective-measurement survival probability instead counts records in which every result is $a$, giving $P_{\mathrm{all}\mathrm{a}}={\cos }^{2n}(\pi /(2n))$. Final return after an earlier departure separates the two finite-$n$ events.

Successively measure in bases whose first vector rotates from $|0⟩$ to $|1⟩$ in $N$ equal angular steps. The probability to remain on that branch throughout is the displayed product of squared overlaps. Without postselection, the final target probability is $[1+{\cos }^N(\pi /N)]/2$. Both approach one for an initial $|0⟩$, but a general input loses its relative phase between the two measurement branches; the process does not implement a coherent Pauli X gate on unknown inputs.