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.

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    What do you mean by C2? – Josu Etxezarreta Martinez Jun 29 at 14:17
  • @JosuEtxezarretaMartinez, C2 is complex 2D space, or the space enough to describe 1 qubit. C2*C2 is for 2 qubits. – Tinkidinki Jun 29 at 14:29
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    I think need to be careful about which symbols you're using for this. I believe the convention is to use a tensor product symbol (\otimes in LaTeX) if you mean the product Hilbert space of two qubits, whereas I would read C_2 x C_2 as the space of 2 x 2 complex matrices, or what is conventionally written as C^{2 x 2}. – SLesslyTall Jun 29 at 16:29
up vote 3 down vote accepted

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.
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    To put a fine point on it, another set of eigenvalues which are meaningful for 4x4 matrices are {-1,+1}, e.g. for the Swap gate, or for a coherent parity measurement (where the latter is a tensor product of Pauli matrices) – Niel de Beaudrap Jul 2 at 8:06
  • @NieldeBeaudrap: I have added your example, and acknowledged you for the credit. – user1271772 Jul 2 at 18:02

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