= Weak value
{title2=$A_w=\langle f|A|i\rangle/\langle f|i\rangle$}
{wiki}
For preselected $|i\rangle$ and nonorthogonal postselected $|f\rangle$, 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 $\langle f|\exp(-igA\otimes p/\hbar)|i\rangle$ to first order gives $\langle f|i\rangle[1-igA_wp/\hbar]$, proving that interpretation within the expansion's validity regime.<Weak values by Yakir Aharonov, David Z. Albert and Lev Vaidman>[1988 primary proposal: weak values in a preselected and postselected spin ensemble]{ref}
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