Weak values and weak coupling maximizing the output of weak measurements

In a weak measurement, the average output $\langle o\rangle$ of a probe that measures an observable $\hat{A}$ of a quantum system undergoing both a preparation 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 for symmetry reasons. This paper answers the questions: what is the weak value that maximizes the output for a fixed coupling? what is the coupling that maximizes the output for a fixed weak value? We derive equations for the optimal values of $A_w$ and $\lambda$, and provide the solutions. The results are independent of the dimensionality of the system, and they apply to a probe having a Hilbert space of arbitrary dimension. Using the Schr\"{o}dinger-Robertson uncertainty relation, we demonstrate that, in an important case, the amplification $\langle o\rangle$ cannot exceed the initial uncertainty $\sigma_o$ in the observable $\hat{o}$, we provide an upper limit for the more general case, and a strategy to obtain $\langle o\rangle\gg \sigma_o$.