Bloch sphere

Definition

Given an orthonormal basis, any pure state | ψ ⟩ of a two-level quantum system can be written as a superposition of the basis vectors | 0 ⟩ and | 1 ⟩ , where the coefficient or amount of each basis vector is a complex number. Since only the relative phase between the coefficients of the two basis vectors has any physical meaning, we can take the coefficient of | 0 ⟩ to be real and non-negative.

We also know from quantum mechanics that the total probability of the system has to be one: ⟨ ψ | ψ ⟩ = 1 , or equivalently | | ψ ⟩ | | 2 = 1 . Given this constraint, we can write | ψ ⟩ using the following representation:

Except in the case where | ψ ⟩ is one of the ket vectors | 0 ⟩ or | 1 ⟩ , the representation is unique. The parameters θ and ϕ , re-interpreted in spherical coordinates as respectively the colatitude with respect to the z-axis and the longitude with respect to the y-axis, specify a point

on the unit sphere in R 3 .

For mixed states, one considers the density operator. Any two-dimensional density operator ρ can be expanded using the identity I and the Hermitian, traceless Pauli matrices σ → :

where a → ∈ R 3 is called the Bloch vector of the system.

It is this vector that indicates the point within the sphere that corresponds to a given mixed state. Specifically, as a basic feature of the Pauli vector, the eigenvalues of ρ are 1 2 ( 1 ± | a → | ) . Density operators must be positive-semidefinite, so we conclude that | a → | ≤ 1 . For pure states, we must then have

in accordance with the above.

As a consequence, the surface of the Bloch sphere represents all the pure states of a two-dimensional quantum system, whereas the interior corresponds to all the mixed states.

u,v,w Representation

The Bloch vector a → = ( u , v , w ) can be represented in the following basis, with reference to the density operator ρ :

where

This basis is often used in laser theory, where w is known as the population inversion.

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 | ψ ⟩ , the isotropy group of | ψ ⟩ , (defined as the set of elements g of U(n) such that g | ψ ⟩ = | ψ ⟩ ) is isomorphic to the product group

In linear algebra terms, this can be justified as follows. Any g of U(n) that leaves | ψ ⟩ invariant must have | ψ ⟩ 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 | ψ ⟩ , 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 2^m.

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

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 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:

Where p i is the probability of the individual states within the ensemble and x i , y i , z i are the coordinates of the individual states (on the surface of Bloch sphere).

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 other (not Bloch-style) parametrizations of (mixed) states that do generalize to dimensions higher than 2.