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If we have a $U$ (unitary with all real entries) such that:
$U|0\rangle =a|0\rangle +b|1\rangle$
What is $U|1\rangle=?$
I know: the definition of what it means to be unitary ie. $U^\dagger U=UU^\dagger =I$
I've worked out: for $U|1\rangle=c|0\rangle+d|1\rangle$ must satisfy $ac+bd=0$ (by taking it's dagger and multiplying it for the constants).
Is this the only information we can derive? How can I write $U|1\rangle$?
You have $U|1\rangle=e^{i\phi}(b^*|0\rangle-a^*|1\rangle)$ (and for real entries, $e^{i\phi}=\pm1$). This condition follows automatically from $$ \langle 0|U^\dagger U |1\rangle=0 $$ -- this is exactly the condition you describe -- together with the fact that $U|0\rangle$ and $U|1\rangle$ must have the same normalization, $$ \langle k|U^\dagger U |k\rangle=1 $$ for $k=0,1$. (This also means that $|a|^2+|b|^2=1$.)
This is the only condition, since now you have ensured that all matrix elements of $U^\dagger U$ are of the correct form.
The general form of a 2x2 unitary matrix is:
$$ \begin{pmatrix} \alpha & \beta \\ -e^{i\phi}\beta^* & e^{i\phi}\alpha^* \end{pmatrix}, $$
with the constraint that $|\alpha|^2 + |\beta|^2$ = 1.
Since you say that $U|0\rangle = U\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} a \\ b \end{pmatrix}$, we have that $\alpha = a$ and $-e^{i\phi}\beta^* = b$.
Therefore, the most we can say about the bottom-right corner is that $d=e^{i\phi}a^*$, and the most we can say about the top-right corner is $c=\beta = -b^* e^{i\phi}$.
So you have: $U|1\rangle = e^{i\phi}b^*|0\rangle - e^{i\phi}a^*|1\rangle$. We therefore do not have enough information to determine the phase $\phi$, but since you only ask how to write $U|1\rangle$ we don't need $\phi$ because it enters our expression for $U|1\rangle$ only as a global phase.
In conclusion: If all we know is $U|0\rangle = a|0\rangle = b|1\rangle$, then we can say that $U|1\rangle = b^*|0\rangle -a^*|1\rangle$, which is correct up to a global phase.
