What are the standard eigenvalues in $\mathbb{C^2}\otimes\mathbb{C^2}$?

Question

In $\mathbb{C^2}$, we generally take $+1$ and $-1$ as the standard eigenvalues, that's what Pauli-X, Pauli-Z measurements, etc will give us. Is there a similar standard while measuring in the Bell basis and the computational basis in $\mathbb{C^2}\otimes\mathbb{C^2}$?

Of course, the actual eigenvalues don't matter, as long as we are talking about the same resolution of identity, but I was just wondering if there was a convention.

Answer 1

There is no notion of "standard eigenvalues" for general matrices.

Some meaningful eigenvalues for 4x4 matrices are:

• {-3/2, -1/2, 1/2, 3/2} which are the possible z-projections of a spin-3/2 particle
• Instead of eigenvalues of X and Z, use the eigenvalues of the Dirac matrices, which are 4x4 matrices that are related to Pauli matrices
• Instead of eigenvalues of X and Z, use the eigenvalues of the 4x4 generalization of the Gell-Mann matrices (which themselves are 3x3 generalizations of the 2x2 Pauli matrices).
• Finally, as Neil de Beaudrap has noted in the comment, {-1,1} can also be eigenvalues for 4x4 matrices, such as the SWAP gate.