Bloch sphere

In quantum mechanics, the Bloch sphere (also known as the Poincaré sphere in optics) is a geometrical representation of the pure state space of a 2-level quantum system. Alternatively, it is the pure state space of a 1 qubit quantum register. The Bloch sphere is actually geometrically a sphere and the correspondence between elements of the Bloch sphere and pure states can be explicitly given.

Bloch sphere

To show this correspondence, consider the qubit description of the Bloch sphere; any pure state ψ can be written as a complex superposition of the ket vectors  |0 \rangle and |1 \rangle ; moreover since global phase factors do not affect physical state, we can take the representation so that the coefficient of  |0 \rangle is real and non-negative. Thus ψ has a representation as

 |\psi \rangle = \cos \theta \, |0 \rangle +  e^{i \phi}  \sin \theta  \,|1 \rangle

with

 -\frac{\pi}{2} \leq \theta < \frac{\pi}{2}, \quad  0 \leq \phi < 2 \pi.

The representation is unique except in the case ψ is one of the ket vectors  |0 \rangle or  |1 \rangle The parameters φ and θ uniquely specify a point on the unit sphere of euclidean space R3, namely the point whose coordinates (x,y,z) are

 \begin{matrix} x & = & \sin 2 \theta \times \cos \phi \\ y & = & \sin 2 \theta \times \sin \phi \\ z & = & \cos 2 \theta \end{matrix}

In this representation  |0 \rangle is mapped into (0,0,1) and  |1 \rangle is mapped into (0,0,-1).

The interior of the Bloch sphere, the open Bloch ball, represents the mixed states of a single qubit. The \vec{r}=(x,y,z) co-ordinates of a state represent the expectation values of the σx,y,z operators respectively. This is conveniently expressed by,


\rho=\frac{1}{2}(\mathbb{I}+\vec{r}\cdot\vec{\sigma})

where \mathbb{I} is the 2x2 identity matrix, and \vec{r}.\vec{\sigma}=\sum_{j=x,y,z}r_j\sigma_j. A convex combination of pure states \{\hat{\vec{r}}_j\} with weights pj gives a mixed state with Bloch vector \vec{r}=\sum_j p_j \hat{\vec{r}}_j.


Generalization

Consider an n-level quantum mechanical system. This system is described by an n-dimensional Hilbert space Hn. The pure state space is by definition the set of 1-dimensional rays of Hn.

Theorem. Let U(n) be the (Lie) group of unitary matrices of size n. Then the pure state space of Hn can be identified to the compact coset space

 \operatorname{U}(n) /(\operatorname{U}(n-1) \times \operatorname{U}(1)).

To prove this fact, note that there is a natural group action of U(n) on the set of states of Hn. This action is continuous and transitive on the pure states. For any state ψ, the fixed point set of ψ, (defined as the set of elements g of U(n) such that g ψ = ψ) is isomorphic to the product group

 \operatorname{U}(n-1) \times \operatorname{U}(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 n2. This is easy to see since the exponential map

 A \mapsto e^{i A}

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 n2.

Corollary. The real dimension of the pure state space of Hn is 2n − 2.

In fact,

 n^2 - ((n-1)^2 +1) = 2 n - 2. \quad

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

Corollary. The real dimension of the pure state space of an m qubit quantum register is 2m+1 − 2.

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. 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, ..., μk with multiplicities n1, ...,nk. Then the group of unitary operators V such that V A V* = A is isomorphic (as a Lie group) to

\operatorname{U}(n_1) \times \cdots \times \operatorname{U}(n_k).

In particular the orbit of A is isomorphic to

\operatorname{U}(n)/(\operatorname{U}(n_1) \times \cdots \times \operatorname{U}(n_k)).