Editing Pauli matrices (section)

=== Some trace relations ===
The following traces can be derived using the commutation and anticommutation relations.

:<math>\begin{align}
  \operatorname{tr}\left(\sigma_j                           \right) &= 0 \\
  \operatorname{tr}\left(\sigma_j \sigma_k                  \right) &= 2\delta_{jk} \\
  \operatorname{tr}\left(\sigma_j \sigma_k \sigma_\ell      \right) &= 2i\varepsilon_{jk\ell} \\
  \operatorname{tr}\left(\sigma_j \sigma_k \sigma_\ell \sigma_m \right) &= 2\left(\delta_{jk}\delta_{\ell m} - \delta_{j\ell}\delta_{km} + \delta_{jm}\delta_{k\ell}\right)
\end{align}</math>

If the matrix {{math|1=''σ''{{sub|0}} = ''I''}} is also considered, these relationships become

<math display=block>\begin{align}
                             \operatorname{tr}\left(\sigma_\alpha  \right)              &= 2\delta_{0 \alpha} \\
                    \operatorname{tr}\left(\sigma_\alpha \sigma_\beta    \right)        &= 2\delta_{\alpha \beta} \\
           \operatorname{tr}\left(\sigma_\alpha   \sigma_\beta   \sigma_\gamma  \right) &= 2 \sum_{(\alpha \beta \gamma)} \delta_{\alpha \beta} \delta_{0 \gamma} - 4 \delta_{0 \alpha} \delta_{0 \beta} \delta_{0 \gamma} + 2i\varepsilon_{0 \alpha \beta \gamma} \\
  \operatorname{tr}\left(\sigma_\alpha \sigma_\beta \sigma_\gamma \sigma_\mu \right)    &= 2\left(\delta_{\alpha \beta}\delta_{\gamma \mu} - \delta_{\alpha \gamma}\delta_{\beta \mu} + \delta_{\alpha \mu}\delta_{\beta \gamma}\right) + 4\left(\delta_{\alpha \gamma} \delta_{0 \beta} \delta_{0 \mu} + \delta_{\beta \mu} \delta_{0 \alpha} \delta_{0 \gamma}\right) - 8 \delta_{0 \alpha} \delta_{0 \beta} \delta_{0 \gamma} \delta_{0 \mu} + 2 i \sum_{(\alpha \beta \gamma \mu)} \varepsilon_{0 \alpha \beta \gamma} \delta_{0 \mu}
\end{align}</math>

where Greek indices {{math|''α'', ''β'', ''γ''}} and {{mvar|μ}} assume values from {{math|{0, ''x'', ''y'', ''z''}<nowiki/>}} and the notation <math display="inline">\sum_{(\alpha \ldots)}</math> is used to denote the sum over the [[cyclic permutation]] of the included indices.