Bloch Sphere - A Generalization For Pure States

A Generalization For Pure States

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 with the compact coset space

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, the isotropy group of, (defined as the set of elements of U(n) such that ) is isomorphic to the product group

In linear algebra terms, this can be justified as follows. Any of U(n) that leaves invariant must have 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, 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 n2. This is easy to see since the exponential map

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,

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.