{scope}

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

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

ID: 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