Exercise 3.11 Discussion

$\def\abs#1{\vert #1 \vert}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 3.11}$

Let $\ket\psi$ be an $n$-qubit state. Show that the sum of the distances from $\ket\psi$ to the standard basis vectors $\ket{j}$ is bounded below by a positive constant that depends only on $n$,

(1)
\begin{align} \sum_j \abs{\ket\psi - \ket{j}} \geq C, \end{align}

where $\abs{\vec{v}}$ indicates the length of the enclosed vector. Specify such a constant $C$ in terms of $n$.