Editing Pauli matrices (section)

=== Completeness relation<span class="anchor" id="completeness_anchor"></span> ===
An alternative notation that is commonly used for the Pauli matrices is to write the vector index {{mvar|k}} in the superscript, and the matrix indices as subscripts, so that the element in row {{mvar|α}} and column {{mvar|β}} of the {{mvar|k}}-th Pauli matrix is {{math|''σ {{sup|k}}{{sub|αβ}}''.}}

In this notation, the ''completeness relation'' for the Pauli matrices can be written
:<math>\vec{\sigma}_{\alpha\beta}\cdot\vec{\sigma}_{\gamma\delta}\equiv \sum_{k=1}^3 \sigma^k_{\alpha\beta}\,\sigma^k_{\gamma\delta} = 2\,\delta_{\alpha\delta} \,\delta_{\beta\gamma} - \delta_{\alpha\beta}\,\delta_{\gamma\delta}.</math>

{{math proof | proof = The fact that the Pauli matrices, along with the identity matrix {{mvar|I}}, form an orthogonal basis for the Hilbert space of all 2 × 2 [[complex number|complex]] matrices <math>\mathcal{M}_{2,2}(\mathbb{C})</math> over <math>\mathbb{C}</math>, means that we can express any  2 × 2 complex matrix {{mvar|M}} as
<math display="block">M = c\,I + \sum_k a_k \,\sigma^k </math>
where {{mvar|c}} is a complex number, and {{mvar|a}} is a 3-component, complex vector. It is straightforward to show, using the properties listed above, that
<math display="block">\operatorname{tr}\left( \sigma^j\,\sigma^k \right) = 2\,\delta_{jk}</math>
where "{{math|tr}}" denotes the [[trace (linear algebra)|trace]], and hence that 
<math display="block">\begin{align}
                   c &={} \tfrac{1}{2}\, \operatorname{tr}\, M\,,\begin{align}&& a_k &= \tfrac{1}{2}\,\operatorname{tr}\,\sigma^k\,M .\end{align} \\[3pt]
  \therefore ~~ 2\,M &= I\,\operatorname{tr}\, M + \sum_k \sigma^k\,\operatorname{tr}\, \sigma^k M ~,
\end{align}</math>
which can be rewritten in terms of matrix indices as
<math display="block">2\, M_{\alpha\beta} = \delta_{\alpha\beta}\,M_{\gamma\gamma} + \sum_k \sigma^k_{\alpha\beta}\,\sigma^k_{\gamma\delta}\,M_{\delta\gamma}~,</math>
where [[Einstein notation|summation over the repeated indices is implied]] {{mvar|γ}} and {{mvar|δ}}. Since this is true for any choice of the matrix {{mvar|M}}, the completeness relation follows as stated above. [[Q.E.D.]]
}}

As noted above, it is common to denote the 2 × 2 unit matrix by {{math|''σ''{{sub|0}},}} so {{math|1=''σ{{sup|0}}{{sub|αβ}}'' = ''δ{{sub|αβ}}''.}} The completeness relation can alternatively be expressed as
<math display="block">\sum_{k=0}^3 \sigma^k_{\alpha\beta}\,\sigma^k_{\gamma\delta} = 2\,\delta_{\alpha\delta}\,\delta_{\beta\gamma} ~ .</math>

The fact that any Hermitian [[complex number|complex]] 2 × 2 matrices can be expressed in terms of the identity matrix and the Pauli matrices also leads to the [[Bloch sphere]] representation of 2 × 2 [[mixed state (physics)|mixed state]]s’ density matrix, ([[Positive semidefinite matrix|positive semidefinite]] 2 × 2 matrices with unit trace. This can be seen by first expressing an arbitrary Hermitian matrix as a real linear combination of {{math|{''σ''{{sub|0}}, ''σ''{{sub|1}}, ''σ''{{sub|2}}, ''σ''{{sub|3}}<nowiki>}</nowiki>}} as above, and then imposing the positive-semidefinite and [[Trace (linear algebra)|trace]] {{math|1}} conditions.

For a pure state, in polar coordinates, <math display="block">\vec{a} = \begin{pmatrix}\sin\theta \cos\phi & \sin\theta \sin\phi & \cos\theta\end{pmatrix},</math> the [[idempotent]] density matrix
<math display="block">
  \tfrac{1}{2} \left(\mathbf{1} + \vec{a} \cdot \vec{\sigma}\right) =
  \begin{pmatrix}
    \cos^2\left(\frac{\,\theta\,}{2}\right) &
      e^{-i\,\phi}\sin\left(\frac{\,\theta\,}{2}\right)\cos\left(\frac{\,\theta\,}{2}\right) \\
    e^{+i\,\phi}\sin\left(\frac{\,\theta\,}{2}\right)\cos\left(\frac{\,\theta\,}{2}\right) &
      \sin^2\left(\frac{\,\theta\,}{2}\right)
  \end{pmatrix}
</math>

acts on the state eigenvector <math>\begin{pmatrix}\cos\left(\frac{\,\theta\,}{2}\right) & e^{+i\phi}\,\sin\left(\frac{\,\theta\,}{2}\right) \end{pmatrix} </math> with eigenvalue +1, hence it acts like a [[projection (linear algebra)|projection operator]].