Bloch Sphere - The Geometry of Density Operators

The Geometry of 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 operators. However, while the Bloch sphere parametrizes not only pure states but mixed states for 2-level systems, 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 μ₁, ..., μ_k with multiplicities n₁, ...,n_k. Then the group of unitary operators V such that V A V* = A is isomorphic (as a Lie group) to

In particular the orbit of A is isomorphic to

We note here that, in the literature, one can find non-Bloch type parametrizations of (mixed) states that do generalize to dimensions higher than 2.