In a weak measurement, the average output of a probe that measures an observable ? of a quantum system undergoing both a preparation in a state ρ_i and a postselection in a state E_f is, to a good approximation, a function of the weak value A_w = Tr[E_f?ρ_i]/ Tr[E_fρ_i], a complex number. For a fixed coupling λ, when the overlap Tr[E_fρ_i] is very small, A_w diverges, but 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 λ, 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?dinger–Robertson uncertainty relation, we demonstrate that, in an important case, the amplification cannot exceed the initial uncertainty σ_o in the observable ?,weprovide an upper limit for the more general case, and a strategy to obtain ? σ_o.
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