Editing Bloch sphere (section)

== Definition ==

Given an orthonormal basis, any [[pure state]] <math>|\psi\rangle</math> of a two-level quantum system can be written as a superposition of the basis vectors <math>|0\rangle</math> and <math>|1\rangle</math>, where the coefficient of (or contribution from) each of the two basis vectors is a [[complex number]]. This means that the state is described by four real numbers. However, only the relative phase between the coefficients of the two basis vectors has any physical meaning (the phase of the quantum system is not directly [[Measurement in quantum mechanics|measurable]]), so that there is redundancy in this description. We can take the coefficient of <math>|0\rangle</math> to be real and non-negative. This allows the state to be described by only three real numbers, giving rise to the three dimensions of the Bloch sphere.

We also know from quantum mechanics that the total probability of the system has to be one:
:<math>\langle\psi | \psi\rangle = 1</math>, or equivalently <math>\big\| |\psi\rangle \big\|^2 = 1</math>.

Given this constraint, we can write <math>|\psi\rangle</math> using the following representation:
:<math>
  |\psi\rangle =
    \cos\left(\theta /2\right) |0 \rangle \, + \, e^{i\phi}  \sin\left(\theta /2\right) |1\rangle =
    \cos\left(\theta /2\right) |0 \rangle \, + \, (\cos\phi + i\sin\phi) \, \sin\left(\theta /2\right) |1\rangle </math>, where <math> 0 \leq \theta \leq \pi</math> and <math>0 \leq \phi < 2 \pi</math>.

The representation is always unique, because, even though the value of <math>\phi</math> is not unique when 
<math>|\psi\rangle</math> is one of the states (see [[Bra-ket notation]]) <math>|0\rangle</math> or <math>|1\rangle</math>, the point represented by <math>\theta</math> and <math>\phi</math> is unique.

The parameters <math>\theta\,</math> and <math>\phi\,</math>, re-interpreted in [[Spherical coordinate system|spherical coordinates]] as respectively the [[colatitude]] with respect to the ''z''-axis and the [[longitude]] with respect to the ''x''-axis, specify a point
:<math>\vec{a} = (\sin\theta \cos\phi,\; \sin\theta \sin\phi,\; \cos\theta) = (u, v, w)</math>

on the unit sphere in <math>\mathbb{R}^3</math>.

For [[mixed state (physics)|mixed state]]s, one considers the [[density operator]].  Any two-dimensional density operator {{mvar|ρ}} can be expanded using the identity {{mvar|I}} and the [[Hermitian matrix|Hermitian]], [[Trace (linear algebra)|traceless]] [[Pauli matrices]] <math>\vec{\sigma}</math>,
:<math>\begin{align}
  \rho &= \frac{1}{2}\left(I + \vec{a} \cdot \vec{\sigma}\right) \\
       &= \frac{1}{2}\begin{pmatrix}
            1 & 0 \\
            0 & 1
          \end{pmatrix} +
          \frac{a_x}{2}\begin{pmatrix}
            0 & 1 \\
            1 & 0
          \end{pmatrix} +
          \frac{a_y}{2}\begin{pmatrix}
            0 & -i \\
            i &  0
          \end{pmatrix} + 
          \frac{a_z}{2}\begin{pmatrix}
            1 &  0 \\
            0 & -1
          \end{pmatrix} \\
       &= \frac{1}{2}\begin{pmatrix}
              1 +  a_z & a_x - ia_y \\
            a_x + ia_y &   1 -  a_z
          \end{pmatrix}
\end{align}</math>,
where <math>\vec{a} \in \mathbb{R}^3</math> is called the '''Bloch vector'''.

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 matrices#Pauli vectors|Pauli vector]], the eigenvalues of {{mvar|ρ}} are <math>\frac{1}{2}\left(1 \pm |\vec{a}|\right)</math>.  Density operators must be positive-semidefinite, so it follows that <math>\left|\vec{a}\right| \le 1</math>.

For pure states, one then has
:<math>\operatorname{tr}\left(\rho^2\right) = \frac{1}{2}\left(1 + \left|\vec{a}\right|^2 \right) = 1 \quad \Leftrightarrow \quad \left|\vec{a}\right| = 1  ~,</math>

in comportance with the above.<ref>The idempotent density matrix
:<math>\frac{1}{2}(1\!\! 1 + \vec a \cdot \vec \sigma) =
  \begin{pmatrix}
    \cos^2 \theta/2                         & \sin \theta/2 ~ \cos\theta/2 ~e^{-i\phi} \\
    \sin \theta/2 ~ \cos\theta/2 ~e^{i\phi} & \sin^2 \theta/2
  \end{pmatrix}
</math>

acts on the state eigenvector <math>(\cos\theta/2, e^{i\phi} \sin\theta/2)</math> with eigenvalue 1, so like a [[Projection (linear algebra)|projection operator]] for it.</ref>

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.