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In a weak measurement, the average output $\langle o\rangle$ of a probe thatmeasures an observable $\hat{A}$ of a quantum system undergoing both apreparation in a state $\rho_i$ and a postselection in a state $E_\mathrm{f}$is, to a good approximation, a function of the weak value $A_w=\mathrm{Tr} [E_f\hat{A} \rho_i]/\mathrm{Tr}[E_f\rho_i]$, a complex number. For a fixed coupling$\lambda$, when the overlap $\mathrm{Tr}[E_f\rho_i]$ is very small, $A_w$diverges, but $\langle o\rangle$ stays finite, often tending to zero forsymmetry reasons. This paper answers the questions: what is the weak value thatmaximizes the output for a fixed coupling? what is the coupling that maximizesthe output for a fixed weak value? We derive equations for the optimal valuesof $A_w$ and $\lambda$, and provide the solutions. The results are independentof the dimensionality of the system, and they apply to a probe having a Hilbertspace of arbitrary dimension. Using the Schr\"{o}dinger-Robertson uncertaintyrelation, we demonstrate that, in an important case, the amplification $\langleo\rangle$ cannot exceed the initial uncertainty $\sigma_o$ in the observable$\hat{o}$, we provide an upper limit for the more general case, and a strategyto obtain $\langle o\rangle\gg \sigma_o$.

Weak values and weak coupling maximizing the output of weak measurements