  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.
