Quantum Zeno Paradox — Measurement Freezing Decay

Survival probability: 1.000

Quantum Zeno Effect

Frequent measurement of an unstable quantum system slows its decay. For a state |ψ⟩ evolving under H, after time δt the survival probability is P = |⟨ψ|e^{−iHδt}|ψ⟩|² ≈ 1 − (ΔH)²(δt)² (quadratic for short times). After n measurements in total time T: τ = T/n → 0, so P^n → 1. The system is "frozen."

Contrast with the Anti-Zeno effect: measurement at resonance frequencies of the environment can accelerate decay. The Zeno/anti-Zeno crossover depends on the spectral density of the bath at the measurement frequency. Demonstrated experimentally: beryllium ions (Itano et al. 1990), cold atoms, NV centers.