Given a complex, two-dimensional vector space, let the vector $ |V\rangle = r_0 e^{i\theta_0} |0\rangle + r_1 e^{i\theta_1} |1\rangle $ correspond to the state of a qubit where $r_0,r_1,\theta_0,\theta_1 \in \mathbb{R}$.
According to p. 22 of Quantum Computing: A Gentle Introduction by Rieffel & Polak, the relative phase of $|V\rangle$ in this standard basis is the unit modulus complex number $e^{iX}$ such that
$$ \frac{r_0e^{i\theta_0}}{r_1e^{i\theta_1}} = e^{iX} \frac{|r_0e^{i\theta_0}|}{|r_1e^{i\theta_1}|} $$
Hence, I derive the relative phase of $|V\rangle$ to be $e^{i(\theta_0-\theta_1)}$ as follows
$$ \frac{r_0e^{i\theta_0}}{r_1e^{i\theta_1}} = e^{iX} \frac{|r_0|}{|r_1|} = e^{iX} \frac{r_0}{r_1} $$
$$ \Leftrightarrow e^{iX} = \frac{r_1}{r_0} \cdot \frac{r_0e^{i\theta_0}}{r_1e^{i\theta_1}} = \frac{e^{i\theta_0}}{e^{i\theta_1}} = e^{i\theta_0-i\theta_1} = e^{i(\theta_0-\theta_1)} $$
However, I have found a few sources stating the relative phase of $|V\rangle$ is defined to be $e^{i(\theta_1-\theta_0)}$, which is the conjugate of what I derived above using the textbook definition.
Can someone help me make sense of this difference? Is there an error in the textbook definition, or am I making some kind of mistake in my calculation?
As I think about it, I believe either answer would be appropriate depending on which modulus one complex number, either $e^{i\theta_0}$ or $e^{i\theta_1}$, you factor out of $|V\rangle$ to distnguish it from all other vectors equal up to global phase. Could someone confirm? Is there a convention on which phase factor one would factor out?
P.S. I did find a related post here, but it doesn't seem to go into sufficient depth to answer my question. Moreover, the author seems to designate the difference $\theta_1 - \theta_0$ or $\theta_0 - \theta_1$ as a relative phase as opposed to the entire phase factor $e^{i(\theta_1-\theta_0)}$ or $e^{i(\theta_0-\theta_1)}$.
Thank you!
The relative phase is... relative ! Concretely, it means that its value depends on which state is considered as the ground/reference state and which one is "the other one".
In the present case, you have : $$ \begin{array}{rclll} |V\rangle &=& r_0e^{i\theta_0}|0\rangle + r_1e^{i\theta_1}|1\rangle \\ &=& e^{i\theta_0}\left(r_0|0\rangle + r_1e^{iX}|1\rangle\right) && (a) \\ &=& e^{i\theta_1}\left(r_0e^{iX}|0\rangle + r_1|1\rangle\right) && (b) \end{array} $$ where (a) $X = \theta_1-\theta_0$ is the relative phase and $\theta_0$ the global phase, whereas (b) $X = \theta_0-\theta_1$ is the relative phase and $\theta_1$ the global phase.
In conlusion, that's a matter of convention. Sometimes, when the convention is too ambiguous, the "relativeness" is explicited by denoting $X_{ij} = \theta_i-\theta_j$ for the relative phase of $|i\rangle$ with respect to the (reference) state $|j\rangle$.