What is the "quantum phase" of a quantum state?

Question

On this page IMBQ docs, until the sentence '..and since the global phase of a quantum state is not detectable..' I follow everything. However 'quantum phase' is introduced without any explaination? What is this?

Answer 1

It is a term of the form $e^{i\phi}$. So, the point is that two states $|\psi\rangle$ and $|\phi\rangle=e^{i\phi}|\psi\rangle$ differ only by this phase. We call it a global phase because it's affecting the whole of the state rather than just part of it.

You can easily see that this has no observable consequences. For example, if we measure in the standard basis, what's the probability of getting the answer $|0\rangle$? $$ p_0=|\langle0|\phi\rangle|^2=|e^{i\phi}\langle0|\psi\rangle|^2=|\langle0|\psi\rangle|^2 $$ The probability is the same for both cases. As it will be for any measurement basis and outcome.