Qubit and conformal mapping

J Lee, C H Kim, E K Lee, S Lee, J Kim

Published Jan 4, 2002• 0 citations

Abstract

Identifying the Bolch sphere with the Riemann sphere(the extended complex plane), we obtain relations between single qubit unitary operations and M\"{o}bius transformations on the extended complex plane.

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OpenAlex

Abstract

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\pi(x_1,x_2,x_3) = \frac{x_1 + i x_2}{1 - x_3}

for sphere

S^2 \to \mathbb{C} \cup \{\infty\}

. Bloch sphere:

|\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle

. Möbius transformation on the Riemann sphere:

z \mapsto \frac{az+b}{cz+d}

ad-bc \neq 0

. Cross-ratio invariance:

\frac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)}

is preserved under Möbius maps. For antipodal points

z

and

-1/\bar{z}

, the cross-ratio with

\{0,1,\infty\}

yields the golden ratio

\varphi = (1+\sqrt{5})/2 \approx 1.618

when

z = e^{i\pi/5}

(or its conjugate), satisfying

z^2 - z - 1 = 0

in the projective coordinate. | CONNECTION: The golden ratio \( \va Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

SU(2)

. We show that the quaternionic stereographic projection intertwines between local unitary group

SU(2)\otimes SU(2)

and quaternionic Möbius transformation. The invariant term appearing in this operation is related to concurrence measure. Yet, there exists the same intertwining stereographic projection for much more global group

Sp(2)

, generalizing the familiar Bloch sphere in 2-level systems. Subsequently, we introduce octonionic stereographic projection and octonionic conformal map (or octonionic Möbius maps) for three-qubit states and find evidence that they may have invariant terms under local unitary operations which shows that both maps are entanglement sensitive.