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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=a0I+a1X+a2Y+a3ZU = a_0I + a_1X + a_2Y +a_3Zο»Ώ.
Also, such a unitary can also be represented this way,
U=[abcd]U = \begin{bmatrix} a & b\\ c & d \end{bmatrix}
and thus, we obtain the following equivalences.
a0=(a+d)/2,Β a1=(b+c)/2,a2=(cβˆ’b)/2i,Β a3=(dβˆ’a)/2a_0 = (a+d)/2,\ a_1 = (b+c)/2,\\ a_2 = (c-b)/2i,\ a_3 = (d-a)/2
Also from UU†=IUU^\dagger = Iο»Ώ we get,
∣a0∣2+∣a1∣2+∣a2∣2+∣a3∣2=1a0βˆ—a1+a1βˆ—a0+ia2βˆ—a3βˆ’ia3βˆ—a2=0a0βˆ—a2βˆ’ia1βˆ—a3+a2βˆ—a0+ia3βˆ—a1=0a0βˆ—a3+ia1βˆ—a2βˆ’ia2βˆ—a1+a3βˆ—a0=0|a_0|^2 + |a_1|^2 + |a_2|^2 + |a_3|^2 = 1\\ a^βˆ—_0a_1+a^βˆ—_1a_0+ia^βˆ—_2a_3βˆ’ia^βˆ—_3a_2 = 0\\ a^βˆ—_0a_2βˆ’ia^βˆ—_1a_3+a^βˆ—_2a_0+ia^βˆ—_3a_1 = 0\\ a^βˆ—_0a_3+ia^βˆ—_1a_2βˆ’ia^βˆ—_2a_1+a^βˆ—_3a_0 = 0
where we define ∣a0∣=cos(θ/2)|a_0| = cos(\theta/2) then ∣a1∣2+∣a2∣2+∣a3∣2=∣sin(θ/2)∣|a_1|^2 + |a_2|^2 + |a_3|^2 = |sin(\theta/2)|.
Then define,
nx=∣a1∣/∣sin(θ/2)∣ny=∣a2∣/∣sin(θ/2)∣nz=∣a3∣/∣sin(θ/2)∣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 nx2+ny2+nz2=1n_x^2 + n_y^2 + n_z^2 = 1ο»Ώ.
Now, we define exp(iΞ±)=a0/cos(ΞΈ/2)exp(iΞ±) =a_0/cos(ΞΈ/2)ο»Ώ and denote the phase of a1,a2,a3a_1,a_2,a_3ο»Ώ as Ξ±1,Ξ±2,Ξ±3Ξ±_1,Ξ±_2,Ξ±_3ο»Ώ respectively.
By putting these in the other constraints we get, Ξ±1=Ξ±2=Ξ±3=Ξ±βˆ’Ο€/2Ξ±_1=Ξ±_2=Ξ±_3=Ξ±βˆ’Ο€/2ο»Ώ.
a0=eiΞ±cos(ΞΈ/2),a1=βˆ’ieiΞ±sin(ΞΈ/2)nx,a2=βˆ’ieiΞ±sin(ΞΈ/2)ny,a3=βˆ’ieiΞ±sin(ΞΈ/2)nza_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=eiΞ±(cos⁑(ΞΈ2)Iβˆ’isin⁑(ΞΈ2)(nxX+nyY+nzZ))=eiΞ±Rn^(ΞΈ)U= e^{iΞ±}(\cos(\frac{ΞΈ}{2})I βˆ’ i\sin(\frac{\theta}{2})(n_xX+n_yY+n_zZ))= e^{iΞ±}R_{\hat n}(ΞΈ)