What is a destructive measurement? What are destructive measurements or incomplete measurements, and what is the (conceptual) difference between them and a usual measurement?
I read somewhere that destructive measurements consume their qubit.

reference:   measurement calculus
p.6 we simplified equations 2 and 3 to equations 4 and 5.
 A: *

*Destructive measurements are processes that completely destroy the system they are measuring, and they are primarily used when detecting light. As an example, to detect the polarization of a photon you can pass it through a polarizing beam splitter and put detectors on either output port: you get complete information of a projective measurement on the photon, but the photon also ceases to exist.
This describes most measurements performed on light, but not all measurements do this, and the alternative, called quantum non-demolition experiments (example, doi), earned Serge Haroche the 2012 Nobel prize in physics.

*An incomplete measurement is a completely different beast: it's a projective measurement that doesn't fully collapse the wavefunction, i.e. it completely collapses some superpositions but it leaves others untouched. As a simple example, suppose that you had the wavefunction
$$ |\psi⟩ = a |1⟩ + b|2⟩ + c|3⟩ + d|4⟩;$$
an example of an incomplete measurement is one that determines whether the state is in either the pair $|1⟩$ and $|2⟩$ or in $|3⟩$ and $|4⟩$, leaving behind the state
$$ |\psi_{12}⟩ = \frac{a |1⟩ + b|2⟩}{\sqrt{|a|^2+|b|^2}}
   \quad\text{or}\quad
   |\psi_{34}⟩ = \frac{c|3⟩ + d|4⟩}{\sqrt{|c|^2+|d|^2}},$$
with probabilities $|a|^2+|b|^2$ and $|c|^2+|d|^2$, respectively. This can be achieved by e.g. making the system interact with some auxiliary two-level system which is later projectively measured, but the details there are not particularly important.
