Editing Bloch sphere (section)

== Pure states ==
Consider an ''n''-level quantum mechanical system.  This system is described by an ''n''-dimensional [[Hilbert space]] ''H''<sub>''n''</sub>.  The pure state space is by definition the set of rays of ''H''<sub>''n''</sub>.

'''Theorem'''.  Let [[U(N)|U(''n'')]] be the [[Lie group]] of unitary matrices of size ''n''.  Then the pure state space of ''H''<sub>''n''</sub> can be identified with the compact coset space
:<math> \operatorname{U}(n) /(\operatorname{U}(n - 1) \times \operatorname{U}(1)). </math>

To prove this fact, note that there is a [[natural transformation|natural]] [[Group action (mathematics)|group action]] of U(''n'') on the set of states of ''H''<sub>''n''</sub>.  This action is continuous and [[transitive group action|transitive]] on the pure states.  For any state <math>|\psi\rangle</math>, the [[isotropy group]] of <math>|\psi\rangle</math>, (defined as the set of elements <math>g</math>  of U(''n'') such that <math>g |\psi\rangle = |\psi\rangle</math>) is isomorphic to the product group

:<math> \operatorname{U}(n - 1) \times \operatorname{U}(1). </math>

In linear algebra terms, this can be justified as follows.  Any <math>g</math> of U(''n'') that leaves <math>|\psi\rangle</math> invariant must have <math>|\psi\rangle</math> as an [[eigenvector]].  Since the corresponding eigenvalue must be a complex number of modulus 1, this gives the U(1) factor of the isotropy group.  The other part of the isotropy group is parametrized by the unitary matrices on the orthogonal complement of <math>|\psi\rangle</math>, which is isomorphic to U(''n'' − 1).  From this the assertion of the theorem follows from basic facts about transitive group actions of compact groups.

The important fact to note above is that the ''unitary group acts transitively'' on pure states.

Now the (real) [[dimension]] of U(''n'') is ''n''<sup>2</sup>.  This is easy to see since the exponential map
:<math> A \mapsto e^{i A} </math>
is a local homeomorphism from the space of self-adjoint complex matrices to U(''n'').  The space of self-adjoint complex matrices has real dimension ''n''<sup>2</sup>.

'''Corollary'''.  The real dimension of the pure state space of ''H''<sub>''n''</sub> is 2''n'' − 2.

In fact,
:<math> n^2 - \left((n - 1)^2 + 1\right) = 2n - 2. \quad </math>

Let us apply this to consider the real dimension of an ''m'' qubit quantum register.  The corresponding Hilbert space has dimension 2<sup>''m''</sup>.

'''Corollary'''.  The real dimension of the pure state space of an ''m''-[[qubit]] [[quantum register]] is 2<sup>''m''+1</sup> − 2.

=== 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>