Ex10 22

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 10.22}$

a) If $\ket\psi$ has persistency $n$ and $\ket\phi$ has persistency $m$, what is the persistency of $\ket\psi\otimes\ket\phi$?

b) Show by induction that the persistency of $\ket{W_n}$ is $n - 1$. (Hint: You may want to use part a).)