1

It's easy to see that in computational basis, Pauli matrices could be represented in the outer product form:

$$ X=|0\rangle\langle1|+|1\rangle\langle0|\\ Y=-i|0\rangle\langle1|+i|1\rangle\langle0|\\ Z=|0\rangle\langle0|-|1\rangle\langle1| $$

If we want to represent the outer products in $X$ basis $|+\rangle$ and $|-\rangle$, one way I can think of is to use the identities $$ |0\rangle=\frac{1}{\sqrt{2}}(|+\rangle+|-\rangle)\\ |1\rangle=\frac{1}{\sqrt{2}}(|+\rangle-|-\rangle) $$ and plug them in the first three equations. I'm wondering is there a simpler / more direct way we can do that? Thanks!

Thanks so much for the answer! For the first method, can I say that (i) the representation is hermitian since it represents a quantum gate? (ii) The trace is 0 because the representation matrix is the Hadamard gate, which is a combination of Pauli matrices? Thanks!! – Zhengrong – 2021-02-18T07:59:32.897

1The representation is hermitian because the operator is self adjoint. Effectively, if it's Hermitian in one representation, it's Hermitian in all representations! (ii) The trace is invariant under representations. Because we're performing a basis transformation $U\sigma U^\dagger$, then $\text{Tr}(U\sigma U^\dagger)=\text{Tr}(\sigma U^\dagger U)=\text{Tr}(\sigma)=0$. – DaftWullie – 2021-02-18T08:52:00.150