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Quantum Information course of the University of Oxford Hilary 2023

Quantum Information course of the University of Oxford Hilary 2023 problem sheet

Quantum Information course of the University of Oxford Hilary 2023 problem sheet 1

{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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