User:AugPi/sandbox

Plotting pure states through stereographic projection

Given a pure state

$\alpha \left|\uparrow \right\rangle +\beta \left|\downarrow \right\rangle =\left|\nearrow \right\rangle$

where $\alpha$ and $\beta$ are complex numbers which are normalized so that

$|\alpha |^{2}+|\beta |^{2}=\alpha ^{*}\alpha +\beta ^{*}\beta =1$

and such that $\langle \downarrow |\uparrow \rangle =0$ and $\langle \downarrow |\downarrow \rangle =\langle \uparrow |\uparrow \rangle =1$, i.e., such that $\left|\uparrow \right\rangle$ and $\left|\downarrow \right\rangle$ form a basis and have diametrically opposite representations on the Bloch sphere, then let

$u={\beta \over \alpha }={\alpha ^{*}\beta \over \alpha ^{*}\alpha }={\alpha ^{*}\beta \over |\alpha |^{2}}=u_{x}+iu_{y}$

be their ratio.

If the Bloch sphere is thought of as being embedded in $\mathbb {R} ^{3}$ with its center at the origin and with radius one, then the plane z = 0 (which intersects the Bloch sphere at a great circle which is the unit circle of plane z = 0) can be thought of as an Argand diagram. Plot point u in this plane — so that in $\mathbb {R} ^{3}$ it has coordinates (x₀, y₀, 0).

Draw a straight line through u and through the point on the sphere that represents $\left|\downarrow \right\rangle$. (Let (0,0,1) represent $\left|\uparrow \right\rangle$ and (0,0,−1) represent $\left|\downarrow \right\rangle$.) This line intersects the sphere at another point besides $\left|\downarrow \right\rangle$. (The only exception is when $u=\infty$, i.e., when a = 0 and $b\neq 0$.) Call this point P. Point u on the plane z = 0 is the stereographic projection of point P on the Bloch sphere. The vector with tail at the origin and tip at P is the direction in 3-D space corresponding to the state $\left|\nearrow \right\rangle$. The coordinates of P are

$P_{x}={2x_{0} \over 1+u_{x}^{2}+u_{y}^{2}}$

$P_{y}={2y_{0} \over 1+u_{x}^{2}+u_{y}^{2}}$

$P_{z}={1-u_{x}^{2}-u_{y}^{2} \over 1+u_{x}^{2}+u_{y}^{2}}$.

Note: mathematically the Bloch sphere can be considered to be a Riemann sphere or a complex 2-dimensional projective Hilbert space, denotable as $\mathbb {P} \mathbf {H} ^{2}$. The complex 2-dimensional Hilbert space $\mathbf {H} ^{2}$ (of which $\mathbb {P} \mathbf {H} ^{2}$ is a projection) is a representation space of SO(3).