 Weak value

For preselected $|i⟩$ and nonorthogonal postselected $|f⟩$, the displayed transition-amplitude ratio controls the first-order conditional pointer response to a von Neumann measurement interaction. It can be complex or outside an observable's eigenvalue range. Expanding $⟨f|\exp (-igA\otimes p/\hslash )|i⟩$ to first order gives $⟨f|i⟩[1-ig{A}_{w}p/\hslash ]$, proving that interpretation within the expansion's validity regime.^[ref]

In quantum mechanics, a "weak value" is a concept that arises in the context of weak measurements, which are a type of measurement that allows observers to extract information about a quantum system with minimal disturbance to the system itself. Weak values are defined in the context of a quantum measurement scenario involving a pre-selected and post-selected ensemble of quantum states.