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The notion of weak measurement in quantum mechanics has gained a significantand wide interest in realizing apparently counterintuitive quantum effects. Inrecent times, several theoretical and experimental works have been reported fordemonstrating the joint weak value of two observables where the couplingstrength is restricted to the second order. In this paper, we extend such aformulation by providing a complete treatment of joint weak measurementscenario 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. Byconsidering the probe state to be discrete as well as continuous variable, wedemonstrate how the joint weak value can be inferred for any given strength ofthe coupling. A particularly interesting result we pointed out that even if theinitial pointer state is uncorrelated, the single pointer displacement canprovide the information about the joint weak value, if at least third order ofthe coupling is taken into account. As an application of our scheme, we providean all-order-coupling treatment of the well-known Hardy paradox by consideringthe continuous as well as discrete meter states and show how the negative jointweak probabilities emerge in the quantum paradoxes at the weak coupling limit.

the european physical journal. d, atomic, molecular and optical physics/the european physical journal. d, atomic, molecular, optical and plasma physics

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