What do the lines above the complex amplitudes in following mean?
$$\{a\left|0\right> + b\left|1\right>,\bar{b}\left|0\right> -\bar{a}\left|1\right>\}$$
I assume it's an orthonormal vector to the first vector, but I'm not sure. They constructed the second term with the first term. How would you construct an orthonormal vector from the first vector?
(These are from a quantum computing book and are quantum states if the context makes it any easier to identify)
It means the complex conjugate.
Let's compute the inner product:
$$(\bar{a}\left<0\right|+\bar{b}\left<1 \right|)(\bar{b}\left|0\right>-\bar{a}\left|1\right>)=\bar{a}\bar{b} \left<0|0\right>-\bar{b}\bar{a}\left< 1|1\right>=0$$