A qubit in |0⟩ undergoes Hamiltonian rotation toward |1⟩. Left alone, it oscillates between the two with period 2π/Ω. Measure it frequently enough, however, and the state freezes — every measurement projects the (still nearly-|0⟩) state right back to |0⟩. In the limit of continuous measurement, the qubit is locked. Watched pots really don't boil.
why measurement freezes evolution
A qubit under Hamiltonian H = (Ω/2)σ_x rotates around the x-axis on the Bloch sphere with frequency Ω. Without measurement, P(|1⟩, t) = sin²(Ωt/2) — Rabi oscillations.
Now measure σ_z at interval τ = 1/γ. After each measurement, the state collapses to whichever outcome occurred. For small τ, the probability of finding |1⟩ is ≈ (Ωτ/2)². Most measurements (probability ≈ 1 − (Ωτ)²/4) find |0⟩ and reset the system. After total time T = N·τ measurements:
P(survived as |0⟩) ≈ (1 − (Ωτ)²/4)^N ≈ exp(−NΩ²τ²/4) = exp(−Ω²·T/(4γ))
As γ → ∞ (continuous measurement), the survival probability tends to 1. The qubit is locked in |0⟩ even though it would normally oscillate. This is the quantum Zeno effect, named for Zeno's paradox of the arrow.
Experiment: confirmed first by Itano et al. (1990) on trapped beryllium ions, then in many platforms (atoms, NV centers, superconducting qubits). It's not a parlor trick — quantum Zeno is used in quantum error correction (frequent error-syndrome measurement freezes decoherence) and in interaction-free measurement (the Elitzur-Vaidman bomb tester).
Slide measurement rate γ: at γ = 0, see the full Rabi oscillation. As γ rises, oscillations decay (Lindblad dynamics). At γ ≫ Ω², the qubit barely moves. Watch quantum mechanics betray its own deterministic evolution under constant observation.