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What are the advantages of working in Pauli basis $(\sigma_0, \sigma_1,\sigma_2, \sigma_3)$, in comparision to the natural basis? Here, $\sigma_0$ is the $2\times2$ identity matrix, and $\sigma_i$ $i=1,2,3$, are the Paul matrices.

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  • $\begingroup$ As opposed to what? $\endgroup$ – Nephente Jul 24 '19 at 11:01
  • $\begingroup$ Say, in comparison to the natural basis. $\endgroup$ – W. Voltera Jul 24 '19 at 13:44
  • $\begingroup$ It manifestly displays cyclic permutability, of use in technical manipulations, of course. $\endgroup$ – Cosmas Zachos Jul 24 '19 at 15:03
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I don't know what you would call the 'natural' basis, but if you're looking for a basis of complex $2\times2$ matrices, the 'Pauli basis' has a bunch of advantages:

  • Basis elements are Hermitean and square to one, convenient for complex conjugates and scalar products.
  • The $\sigma_i$ are traceless, again convenient for scalar products and projections.
  • The basis is closed under commutation and anticommutation, convenient for Lie algebra stuff. Expressions in terms of $\epsilon_{ijk}$ and $\delta_{ij}$ are simple.
  • It is standard, convenient for looking up identities.
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