Weak values and weak coupling maximizing the output of weak measurements

• We have provided a general framework to find the extremal values of a weak measurement.
• We have derived the location of the extremal values in terms of preparation and postselection.
• We have devised a maximization strategy going beyond the limit of the Schrödinger–Robertson relation.

In a weak measurement, the average output 〈o〉〈o〉 of a probe that measures an observable Aˆ of a quantum system undergoing both a preparation in a state ρi and a postselection in a state Ef is, to a good approximation, a function of the weak value Aw=Tr[EfAˆρi]/Tr[Efρi], a complex number. For a fixed coupling λλ, when the overlap Tr[Efρi] is very small, AwAw diverges, but 〈o〉〈o〉 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 AwAw and λλ, 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 Schrödinger–Robertson uncertainty relation, we demonstrate that, in an important case, the amplification 〈o〉〈o〉 cannot exceed the initial uncertainty σoσo in the observable oˆ, we provide an upper limit for the more general case, and a strategy to obtain 〈o〉≫σo〈o〉≫σo.