# What are the advantages of working in Pauli basis? [closed]

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.

• As opposed to what? – Nephente Jul 24 '19 at 11:01
• Say, in comparison to the natural basis. – W. Voltera Jul 24 '19 at 13:44
• It manifestly displays cyclic permutability, of use in technical manipulations, of course. – Cosmas Zachos Jul 24 '19 at 15:03

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:
• 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.