Exercise 2.13

$\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 2.13}$

Relate the four parametrizations of the state space of a single qubit to each other: Give formulas for

a) vectors in ket notation

b) elements of the extended complex plane

c) spherical coordinates for the Bloch sphere (see exercise 2.12) in terms of the $x$, $y$, and $z$ coordinates of the Bloch sphere.

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a) $\alpha \vert 0 \rangle + \beta \vert 1 \rangle = \begin{bmatrix} \alpha\\\beta \end{bmatrix} = \begin{bmatrix} \alpha & \beta \end{bmatrix}^{T} = \alpha \vert 0 \rangle + \beta \vert 1 \rangle$

b) $a \vert 0 \rangle + b \vert 1 \rangle \to \frac{b}{a}=\alpha$, $\alpha \to \frac{1}{\sqrt{1+\vert \alpha \vert ^2}} \vert 0 \rangle + \frac{\alpha}{\sqrt{1+ \vert \alpha \vert ^2}}$, and $\infty ↔ 1$

c) $(x, y, z) = (sin(\theta) cos(\phi), sin(\theta) sin(\phi), cos(\theta))$

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