5
$\begingroup$

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.

|improve this question
$\endgroup$
  • 1
    $\begingroup$ What do you mean by C2? $\endgroup$ – Josu Etxezarreta Martinez Jun 29 '18 at 14:17
  • $\begingroup$ @JosuEtxezarretaMartinez, C2 is complex 2D space, or the space enough to describe 1 qubit. C2*C2 is for 2 qubits. $\endgroup$ – Mahathi Vempati Jun 29 '18 at 14:29
  • 1
    $\begingroup$ 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}. $\endgroup$ – SLesslyTall Jun 29 '18 at 16:29
3
$\begingroup$

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.
|improve this answer
$\endgroup$
  • 1
    $\begingroup$ 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) $\endgroup$ – Niel de Beaudrap Jul 2 '18 at 8:06
  • $\begingroup$ @NieldeBeaudrap: I have added your example, and acknowledged you for the credit. $\endgroup$ – user1271772 Jul 2 '18 at 18:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.