Editing Bloch sphere (section)

== Density operators ==
Formulations of quantum mechanics in terms of pure states are adequate for isolated systems; in general quantum mechanical systems need to be described in terms of [[density matrix|density operators]].  The Bloch sphere parametrizes not only pure states but mixed states for 2-level systems.  The density operator describing the mixed-state of a 2-level quantum system (qubit) corresponds to a point ''inside'' the Bloch sphere with the following coordinates:
:<math> \left( \sum p_i x_i,  \sum p_i y_i,  \sum p_i z_i \right),</math>

where <math>p_i</math> is the probability of the individual states within the ensemble and <math>x_i, y_i, z_i</math> are the coordinates of the individual states (on the ''surface'' of Bloch sphere).  The set of all points on and inside the Bloch sphere is known as the ''Bloch ball.''

For states of higher dimensions there is difficulty in extending this to mixed states.  The topological description is complicated by the fact that the unitary group does not act transitively on density operators.  The orbits moreover are extremely diverse as follows from the following observation:

'''Theorem'''. Suppose ''A'' is a density operator on an ''n'' level quantum mechanical system whose distinct eigenvalues are μ<sub>1</sub>, ..., μ<sub>''k''</sub> with multiplicities ''n''<sub>1</sub>, ..., ''n''<sub>''k''</sub>.  Then the group of unitary operators ''V'' such that ''V A V''* = ''A'' is isomorphic (as a Lie group) to
:<math>\operatorname{U}(n_1) \times \cdots \times \operatorname{U}(n_k).</math>

In particular the orbit of ''A'' is isomorphic to
:<math>\operatorname{U}(n)/\left(\operatorname{U}(n_1) \times \cdots \times \operatorname{U}(n_k)\right).</math>

It is possible to generalize the construction of the Bloch ball to dimensions larger than 2, but the geometry of such a "Bloch body" is more complicated than that of a ball.{{sfn | Appleby | 2007 }}