Pauli Matrices -- Properties

Some Useful Properties of the Pauli Matrices
- anticommutation relations
- commutation relations
- determinant and trace
- inverse
- hermiticity

[ review of summation, Kroneker delta, Levi-Civita symbol, etc. ]

--
Prof. Dr. Ricardo Gomes (IF - UFG)

[MrVsoral]

Hello Prof, Ricardo

I watched your video on Pauli Matrices as I was searching for answer to one question. Hope you would respond and clarify.

Consider a quantum system comprising of electron spins.

We know that the Pauli matrices X and Z do not commute => X.Z ≠ Z.X
This means that we cannot simultaneously measure Spin along X and Z directions of an electron.

However if we consider the operators (Z × Z) and (X × X) which are formed by tensor products of Z's and X's; then these operators do commute;
i.e. (Z × Z)(X × X) = (X × X)(Z × Z).


If we consider a system of two electrons, then -
(Z × Z) is equivalent of measuring Z observable (i.e. spin along Z direction) on two electrons,
and (X × X) is similarly equivalent of measuring X observable (i.e. spin along X direction) on two electrons.
If this interpretation is correct, then logically it seems counterintuitive that if we take a system of two electrons, then we can simultaneously measure spins along Z and X directions for both!