weak value

A weak value is the quantity Aw = ⟨ϕ|A|ψ⟩/⟨ϕ|ψ⟩ obtained from a weak — minimally disturbing — measurement of an observable A on a quantum system pre-selected into state |ψ⟩ and later post-selected into state |ϕ⟩; because the denominator can be small, a weak value can lie outside A's normal eigenvalue range. Introduced by Aharonov, Albert & Vaidman in 1988, it is the formal device behind both the original "spin can turn out to be 100" result and the 2016 quantum pigeonhole principle, and underlies the two-state vector formalism that Yakir Aharonov and Vaidman built around it.

References

• claim-aharonov-1988-weak-value-origin-disturbance-scales-epsilon-squared

• claim-aharonov-2016-quantum-pigeonhole-three-particles-two-boxes-weak-value

• claim-lloyd-et-al-2010-p-ctc-paper-cites-aharonov-weak-value-and-aaronson-postselection

• 2026-10-28: a 2010 physics paper on postselected closed timelike curves (P-CTCs) cites the 1988 weak-value paper directly to describe why P-CTCs lack a unique definite state at a given time — the weak-value concept's one documented reach into the physics-of-computation literature outside its originating foundations-of-QM context (claim-lloyd-et-al-2010-p-ctc-paper-cites-aharonov-weak-value-and-aaronson-postselection).