Source: cirosantilli/quantum-information-course-of-the-university-of-oxford-hilary-2023/quantum-information-course-of-the-university-of-oxford-hilary-2023-problem-sheet-1/1/a

= a
{scope}

It is the <norm induced by the complex dot product> over <\C^2>:
$$
|\ket{\psi}|
= \sqrt{\left|\frac{1 + i}{2}\right|^2 + \left|\frac{1-i}{2}\right|^2}
= \sqrt{\left|\frac{1}{2} + i\frac{1}{2}\right|^2 + \left|\frac{1}{2} + i\frac{-i}{2}\right|^2}
= \sqrt{
    \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^2}^2 +
    \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{-1}{2}\right)^2}^2
  }
= \sqrt{
    \left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^2 +
    \left(\frac{1}{2}\right)^2 + \left(\frac{-1}{2}\right)^2
  }
= \sqrt{\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}} +
= \sqrt{\frac{1 + 1 + 1 + 1}{4}}
= 1
$$

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