Exa 6

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ A.6}$

a) Give an example of three density operators no two of which can be simultaneously diagonalized in that there does not exist a basis with respect to which both are diagonal.

b) Show that if a set of density operators commute then they can be simultaneously diagonalized.