Weak values and weak coupling maximizing the output of weak measurements Original Research Article

In a weak measurement, the average output image of a probe that measures an observable image of a quantum system undergoing both a preparation in a state image and a postselection in a state image is, to a good approximation, a function of the weak value image, a complex number. For a fixed coupling image, when the overlap image is very small, image diverges, but image 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 image and image, 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 image cannot exceed the initial uncertainty image in the observable image, we provide an upper limit for the more general case, and a strategy to obtain image.