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We can express the square of the “length” of our quantum state vector in $\ket S = a_\text{h} \ket{\text{hlp}} + a_\text{v} \ket{\text{vlp}}$ as follows:

$$\Vert~\vert S\rangle~\Vert^2 = a_{\text{h}}^2 + a_{\text{v}}^2.$$

I used quote marks around length for a state vector because that length is not something you can measure with a ruler. For almost all our work in quantum computing, we will use state vectors whose length is $1$ (they are called “unit vectors” or “normalized vectors”). Following that convention, we will insist that

$$a_{\text{h}}^2 + a_{\text{v}}^2 =1.$$

We will motivate that requirement in the next chapter.