Chemistry - Bloch sphere

Bloch sphere

In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a 2-level quantum mechanical 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.

To show this correspondence, consider the qubit description of the Bloch sphere; any state ψ can be written as a complex superposition of the ket vectors $|0 \rangle$ and $|1 \rangle$ ; moreover since 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).

Generalization

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

Theorem. Let U(n) be the (Lie) group of unitary matrices of size n. Then the pure state space of H_n 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 H_n. 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 elemenrs 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 H_n 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 2^m.

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

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

In particular the orbit of A is isomorphic to

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