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Proof that Bloch Sphere Unitaries as Rotations
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Proof that Bloch Sphere Unitaries as Rotations
A single qubit operator can be represented as $U = a_0I + a_1X + a_2Y +a_3Z$ο»Ώ.
Also, such a unitary can also be represented this way,
$U = \begin{bmatrix} a & b\\ c & d \end{bmatrix}$
and thus, we obtain the following equivalences.
$a_0 = (a+d)/2,\ a_1 = (b+c)/2,\\ a_2 = (c-b)/2i,\ a_3 = (d-a)/2$
Also from $UU^\dagger = I$ο»Ώ we get,
$|a_0|^2 + |a_1|^2 + |a_2|^2 + |a_3|^2 = 1\\ a^β_0a_1+a^β_1a_0+ia^β_2a_3βia^β_3a_2 = 0\\ a^β_0a_2βia^β_1a_3+a^β_2a_0+ia^β_3a_1 = 0\\ a^β_0a_3+ia^β_1a_2βia^β_2a_1+a^β_3a_0 = 0$
where we define $|a_0| = cos(\theta/2)$ο»Ώ then $|a_1|^2 + |a_2|^2 + |a_3|^2 = |sin(\theta/2)|$ο»Ώ.
Then define,
$n_x = |a_1|/|sin(\theta/2)|\\ n_y = |a_2|/|sin(\theta/2)|\\ n_z = |a_3|/|sin(\theta/2)|$
and further we get $n_x^2 + n_y^2 + n_z^2 = 1$ο»Ώ.
Now, we define $exp(iΞ±) =a_0/cos(ΞΈ/2)$ο»Ώ and denote the phase of $a_1,a_2,a_3$ο»Ώ as $Ξ±_1,Ξ±_2,Ξ±_3$ο»Ώ respectively.
By putting these in the other constraints we get, $Ξ±_1=Ξ±_2=Ξ±_3=Ξ±βΟ/2$ο»Ώ.
$a_0 = e^{i\alpha}cos(\theta/2),\\ a_1 = -ie^{i\alpha}sin(\theta/2)n_x,\\ a_2 = -ie^{i\alpha}sin(\theta/2)n_y,\\ a_3 = -ie^{i\alpha}sin(\theta/2)n_z$
$U= e^{iΞ±}(\cos(\frac{ΞΈ}{2})I β i\sin(\frac{\theta}{2})(n_xX+n_yY+n_zZ))= e^{iΞ±}R_{\hat n}(ΞΈ)$