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.