Exercise 3.9 Discussion

$\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 3.9}$

a) Show that any $n$-qubit quantum state can be represented by a vector of the form

(1)
\begin{align} a_0\ket{0\dots 00} + a_1\ket{0\dots 01} + \dots+a_{2^n-1}\ket{1\dots 11} \end{align}

where the first non-zero $a_i$ is real and non-negative.
b) Show that this representation is unique in the sense that any two different vectors of this form represent different quantum states.