How do two qubit states differing by a global phase relate to each other?

Question

I have looked at the following:

What is the difference between a relative phase and a global phase? In particular, what is a phase?

Global and relative phases of kets in QM

Global phases and indistinguishable quantum states, mathematical understanding

If two states differ by a scalar of magnitude of 1, then they are indistinguishable. Consider: \begin{align} \vert \psi_1 \rangle &= \dfrac{1}{\sqrt{2}} \vert 0 \rangle + \dfrac{i}{\sqrt{2}} \vert 1 \rangle\\ \vert \psi_2 \rangle &= \color{red}{i}\left(\dfrac{-i}{\sqrt{2}} \vert 0 \rangle + \dfrac{1}{\sqrt{2}} \vert 1 \rangle\right). \end{align}

Which of the following is true about $\vert \psi_1 \rangle$ and $\vert \psi_2 \rangle$?

1. $\vert \psi_1 \rangle = \vert \psi_2 \rangle$
2. $\vert \psi_1 \rangle \neq \dfrac{-i}{\sqrt{2}} \vert 0 \rangle + \dfrac{1}{\sqrt{2}} \vert 1 \rangle$
3. $\vert \psi_1 \rangle = \dfrac{-i}{\sqrt{2}} \vert 0 \rangle + \dfrac{1}{\sqrt{2}} \vert 1 \rangle$ up to global phase.
4. If we just ignore the global phase in $\vert \psi_2 \rangle$ and only deal with $\dfrac{-i}{\sqrt{2}} \vert 0 \rangle + \dfrac{1}{\sqrt{2}} \vert 1 \rangle$ , do we still have the state vector on a Bloch sphere yields the same projection as $\vert \psi_1 \rangle$?

Lastly, since the global phase is not physically observable, is it mathematically evident?

Answer 1

1) So $|\psi_1\rangle \neq |\psi_2\rangle$, but it effectively is since they give the exact same distributions for any measurement in any basis.

2) Same discussion as above.

3) True

4) States in the Bloch sphere are of the form

$$|\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle,$$

so the state you describe technically is not directly on the Bloch sphere. I think a better way of thinking about global phase is that it's an infinite equivalence class of states with the exact same physical properties, and one representative (the one with a real coefficient in front of $|0\rangle$) is on the Bloch sphere.

You cannot measure the global phase. Phases are only relevant when they are relative, and consequently affect superpositions/measurements.