Weak values and weak coupling maximizing the output of weak measurements

Abstract

In a weak measurement, the average output 〈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 λ, when the overlap Tr[Efρi] is very small, Aw diverges, but 〈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 Aw 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 Schrödinger–Robertson uncertainty relation, we demonstrate that, in an important case, the amplification 〈o〉 cannot exceed the initial uncertainty σo in the observable oˆ, we provide an upper limit for the more general case, and a strategy to obtain 〈o〉≫σo.