Joint weak value for all order coupling using continuous variable and qubit probe

Abstract

The notion of weak measurement in quantum mechanics has gained a significant and wide interest in realizing apparently counterintuitive quantum effects. In recent times, several theoretical and experimental works have been reported for demonstrating the joint weak value of two observables where the coupling strength is restricted to the second order. In this paper, we extend such a formulation by providing a complete treatment of joint weak measurement scenario for all-order-coupling for the observable satisfying $A^2=\mathbb{I}$ and $A^2=A$, which allows us to reveal several hitherto unexplored features. By considering the probe state to be discrete as well as continuous variable, we demonstrate how the joint weak value can be inferred for any given strength of the coupling. A particularly interesting result we pointed out that even if the initial pointer state is uncorrelated, the single pointer displacement can provide the information about the joint weak value, if at least third order of the coupling is taken into account. As an application of our scheme, we provide an all-order-coupling treatment of the well-known Hardy paradox by considering the continuous as well as discrete meter states and show how the negative joint weak probabilities emerge in the quantum paradoxes at the weak coupling limit.