Re-writing in sign basis.

Question

$\newcommand\ket[1]{\left\vert #1\right\rangle}$ Let $\ket\phi = 12 \ket{0} + 1 + 2\sqrt{i2}\ket{1}$. Write $\ket\phi$ in the form $\alpha_0\ket{+} + \alpha_1\ket{-}$. What is $\alpha_0$?

I came across this problem in a course i am doing, i have been struggling writing things in sign basis, much appreciated.

Answer 1

The "sign basis" is defined as $$\begin{aligned} \lvert+\rangle &= \frac{1}{\sqrt{2}}\left(\lvert0\rangle+\lvert1\rangle\right)\\ \lvert-\rangle &= \frac{1}{\sqrt{2}}\left(\lvert0\rangle-\lvert1\rangle\right) \end{aligned}$$ Those equations can be solved for $\lvert0\rangle$ and $\lvert1\rangle$. Then it's just a matter of inserting in the given state and comparing the coefficients.

Answer 2

Hint: All you need to know is that $|0>=\frac{1}{\sqrt{2}}(|+> +|->)$ and $|1>=\frac{1}{\sqrt{2}}(|+>-|->).$