Weak values and weak coupling maximizing the output of weak measurements

TL;DR

This paper derives optimal conditions for weak values and couplings to maximize measurement output in quantum weak measurements, providing bounds and strategies for amplification beyond initial uncertainties.

Contribution

It introduces equations for optimal weak values and couplings, offering a general solution applicable to arbitrary system dimensions and measurement observables.

Findings

Optimal weak value and coupling equations derived

Maximum amplification limited by initial observable uncertainty

Strategies proposed to surpass initial uncertainty bounds

Abstract

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 $\langle o\rangle\gg \sigma_o$.