  Unlike the real part of the generalized weak value of an observable, which
can in a restricted sense be operationally interpreted as an idealized
conditioned average of that observable in the limit of zero measurement
disturbance, the imaginary part of the generalized weak value does not provide
information pertaining to the observable being measured. What it does provide
is direct information about how the initial state would be unitarily disturbed
by the observable operator. Specifically, we provide an operational
interpretation for the imaginary part of the generalized weak value as the
logarithmic directional derivative of the post-selection probability along the
unitary flow generated by the action of the observable operator. To obtain this
interpretation, we revisit the standard von Neumann measurement protocol for
obtaining the real and imaginary parts of the weak value and solve it exactly
for arbitrary initial states and post-selections using the quantum operations
formalism, which allows us to understand in detail how each part of the
generalized weak value arises in the linear response regime. We also provide
exact treatments of qubit measurements and Gaussian detectors as illustrative
special cases, and show that the measurement disturbance from a Gaussian
detector is purely decohering in the Lindblad sense, which allows the shifts
for a Gaussian detector to be completely understood for any coupling strength
in terms of a single complex weak value that involves the decohered initial
state.
