Quantum mechanics is based on the idea of waves, and waves have both a magnitude and a phase? For example, in $$|\psi\rangle = i\alpha|0\rangle + \beta|1\rangle.$$ does $\alpha$ and $\beta$ represent magnitude and $i$ represents a phase? Or how do we represent phase? Or is it something else?
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It is more common to write a qubit as
$$ |\psi\rangle = \alpha|0\rangle + \beta|1\rangle, $$
where $\alpha, \beta \in \mathbb{C}$, i.e. to omit $i$ and get it in complex number $\alpha$.
Parameters $\alpha$ and $\beta$ are called complex amplitudes, $|\alpha|^2$ is a probability of measuring state $|0\rangle$ and $|\beta|^2$ is a probability of measuring state $|1\rangle$. So, $\alpha$ and $\beta$ (or rather square of their absolute values) can be called "magnitude".
Any qubit can be rewritten as
$$ |\psi\rangle = \cos(\theta/2)|0\rangle + \mathrm{e}^{i\phi}\sin(\theta/2)|1\rangle, $$
where $\theta$ and $\phi$ are coordinates on so-called Bloch sphere.
In this notation, $\cos^2(\theta/2)$ and $\sin^2(\theta/2)$ are probabilities ("magnitudes") of measuring $|0\rangle$ and $|1\rangle$, respecitively. Parameter $\phi$ is a phase (or to be precise, relative phase).
Martin Vesely
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