# Worked Problems in Physics

## Nielsen and Chuang Exercise 2.62

Posted in Nielsen/Chuang by rpmuller on March 25, 2010

Exercise 2.62 Show that any measurement where the measurement operators and the POVM elements coincide is a projective measurement.

The POVM elements satisfy ${\sum_m E_m=I}$. The measurement operators satisfy ${\sum_mM^\dagger_mM_m=I}$. For these two operators to be the same, ${E_m=M_m}$, which requires that ${M^\dagger_mM_m=M_m}$. This is close to the definition of a projector, which is that ${M^2=M}$. For the final component, we must show that the matrix ${M_m}$ is Hermitian, which it is if it is equivalent to the POVM elements. QED.