Weak measurement with orthogonal pre-selection and post-selection

Weak measurement is a novel quantum measurement scheme, which is usually characterized by the weak value formalism. To guarantee the validity of the weak value formalism, the fidelity between the pre-selection and the post-selection should not be too small generally. In this work, we study the weak measurement on a qubit system with exactly or asymptotically orthogonal pre- and post-selections. We shall establish a general rigorous framework for the weak measurement beyond the weak value formalism, and obtain the average output of a weak measurement when the pre- and post-selections are exactly orthogonal. We shall also study the asymptotic behavior of a weak measurement in the limiting process that the pre- and post-selections tend to be orthogonal.