# 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.

• 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.