Editing Bloch sphere (section)

=== Plotting pure two-spinor states through stereographic projection ===
[[File:Riemann Spin2States.jpg|thumb|upright=1.3|Bloch sphere centered at the origin of <math>\mathbb{R}^3</math>. A pair of points on it, <math>\left|\uparrow\right\rangle</math> and <math>\left|\downarrow\right\rangle</math> have been chosen as a basis. Mathematically they are orthogonal even though graphically the angle between them is &pi;. In <math>\mathbb{R}^3</math> those points have coordinates (0,0,1) and (0,0,&minus;1). An arbitrary [[spinor]] <math>\left|\nearrow\right\rangle</math> on the Bloch sphere is representable as a unique linear combination of the two basis spinors, with coefficients being a pair of complex numbers; call them ''&alpha;'' and ''&beta;''. Let their ratio be <math>u = {\beta \over \alpha}</math>, which is also a complex number <math>u_x + i u_y</math>. Consider the plane ''z''&nbsp;=&nbsp;0, the equatorial plane of the sphere, as it were, to be a complex plane and that the point ''u'' is plotted on it as <math>(u_x, u_y, 0)</math>. Project point ''u'' stereographically onto the Bloch sphere away from the South Pole — as it were — (0,0,&minus;1). The projection is onto a point marked on the sphere as <math>\left|\nearrow\right\rangle</math>.]]
Mathematically the Bloch sphere for a two-spinor state can be mapped to a [[Riemann sphere]] <math>\mathbb{C}\mathbf{P}^1</math>, i.e., the [[projective Hilbert space]] <math>\mathbf{P}(H_2)</math> with the 2-dimensional complex Hilbert space <math>H_2</math> a [[Representation_theory|representation space]] of [[SO(3)]].{{sfn | Penrose | 2007 | p=554}}
Given a pure state 
: <math> \alpha \left|\uparrow \right\rangle + \beta \left|\downarrow \right\rangle = \left|\nearrow \right\rangle </math>
where <math>\alpha</math> and <math>\beta</math> are complex numbers which are normalized so that 
: <math> |\alpha|^2 + |\beta|^2 = \alpha^* \alpha + \beta^* \beta = 1</math>
and such that <math>\langle\downarrow | \uparrow\rangle = 0</math> and <math>\langle\downarrow | \downarrow\rangle = \langle\uparrow | \uparrow\rangle = 1</math>,
i.e., such that <math>\left|\uparrow\right\rangle</math> and <math>\left|\downarrow\right\rangle</math> form a basis and have diametrically opposite representations on the Bloch sphere, then let
:<math> u = {\beta \over \alpha} = {\alpha^* \beta \over \alpha^* \alpha} = {\alpha^* \beta \over |\alpha|^2} = u_x + i u_y</math>
be their ratio.

If the Bloch sphere is thought of as being embedded in <math>\mathbb{R}^3</math> with its center at the origin and with radius one, then the plane ''z''&nbsp;=&nbsp;0 (which intersects the Bloch sphere at a great circle; the sphere's equator, as it were) can be thought of as an [[Argand diagram]]. Plot point ''u'' in this plane — so that in <math>\mathbb{R}^3</math> it has coordinates <math>(u_x, u_y, 0)</math>.

Draw a straight line through ''u'' and through the point on the sphere that represents <math>\left|\downarrow\right\rangle</math>. (Let (0,0,1) represent <math>\left|\uparrow\right\rangle</math> and (0,0,&minus;1) represent <math>\left|\downarrow\right\rangle</math>.) This line intersects the sphere at another point besides <math>\left|\downarrow\right\rangle</math>. (The only exception is when <math>u = \infty</math>, i.e., when <math>\alpha = 0</math> and <math>\beta \ne 0</math>.) 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 spinor <math>\left|\nearrow\right\rangle</math>. The coordinates of ''P'' are
:<math> P_x = {2 u_x \over 1 + u_x^2 + u_y^2},</math>
:<math>P_y = {2 u_y \over 1 + u_x^2 + u_y^2},</math>
:<math>P_z = {1 - u_x^2 - u_y^2 \over 1 + u_x^2 + u_y^2}.</math>