Weak Values & Quantum Back-Action

Aharonov-Albert-Vaidman protocol (1988)

A_w = ⟨ψ_f|A|ψ_i⟩ / ⟨ψ_f|ψ_i⟩

Pointer shift:
⟨q⟩_f = g·Re(A_w)
⟨p⟩_f = g·Im(A_w)

Strong: collapse |ψ⟩→|a⟩
Weak: tiny disturbance

Pre-select |ψᵢ⟩ angle θᵢ (°) 45

Post-select |ψ_f⟩ angle θ_f (°) 135

Coupling g (0=weak, 1=strong) 0.1

Pointer width σ 2.0

Aᵥᵥ = —

Weak measurement: couple system to pointer weakly (g≪σ). Post-select on outcome. The pointer shifts by the weak value A_w, which can be outside the eigenvalue range — anomalous weak values signal quantum interference. As g→strong, weak value collapses to eigenvalue statistics.