Quantum Measurement: Weak Value Amplification

A_w = ⟨f|A|i⟩/⟨f|i⟩ — weak values can exceed the eigenvalue spectrum, enabling sub-shot-noise precision.

Pre-select |i⟩ angle θᵢ 45°

Post-select |f⟩ angle θ_f 92°

Coupling g 0.10

Meter σ 2.0

Aharonov-Albert-Vaidman (1988): pre-select state |i⟩, weakly couple to meter, post-select |f⟩. The meter shifts by g·Re(A_w). Near-orthogonal pre/post-selection gives |A_w| ≫ eigenvalues. Used to amplify tiny signals (spin Hall effect of light, beam deflections of order ångströms).