Aharonov's 1988 weak-value concept originated from noticing measurement disturbance scales as epsilon-squared while extracted information scales as epsilon

physicsquantum-foundationsweak-valueweak-measurementhistory-of-science

Yakir Aharonov describes the origin of weak measurement as starting from a spin-1/2 thought experiment: measure σx, then σy, and ask what a measurement of σx+σy would show. He first expected the sum to equal 2, then realized a direct measurement of σx+σy necessarily disturbs both σx and σy. This led him to consider a measurement that interacts only weakly with the particle — one whose extracted information scales as the coupling strength ϵ while the disturbance to the system scales as ϵ², so that over an ensemble of order 1/ϵ particles the signal becomes visible despite the weak coupling per particle. This mechanism underlies "How the Result of a Measurement of a Component of the Spin of a Spin-1/2 Particle Can Turn Out to be 100" (Aharonov, Albert, and Vaidman, 1988, Physical Review Letters 60:1351), the paper that introduced the weak value Aw = ⟨ϕ|A|ψ⟩/⟨ϕ|ψ⟩ for pre-selected |ψ⟩ and post-selected |ϕ⟩ — a quantity that can exceed the eigenvalue range of A because the denominator can be small.