The Even Prime
The question in dimension 2 is, when defining the circulant MUB operator 
why do we need the complex number $i$? Why couldn't we just take the circulant $C = (1,-1)$? Well, $C$ has the obvious problem that both eigenvector columns are the same, making it useless as a basis. But let's step back and think a moment about a general circulant of the form $(1, \omega)$, where we don't know exactly what $\omega$ should be. Then conjugating on the diagonal $\sigma_{Z}$, which lists the usual spin eigenvalues, results in 
Now checking the independence of eigenvectors in the relation 
tells us that $\omega^{2} \neq 1$, namely that $-1$ is not an option. Then the eigenvalue equation 
tells us that $(\overline{\omega}^{2} - 1) \overline{\omega} = \omega - \overline{\omega}$, which is to say that $\overline{\omega}^{3} = \omega$. So we simply must have that $\omega = i$, if it's an ordinary number of some sort.
why do we need the complex number $i$? Why couldn't we just take the circulant $C = (1,-1)$? Well, $C$ has the obvious problem that both eigenvector columns are the same, making it useless as a basis. But let's step back and think a moment about a general circulant of the form $(1, \omega)$, where we don't know exactly what $\omega$ should be. Then conjugating on the diagonal $\sigma_{Z}$, which lists the usual spin eigenvalues, results in 
Now checking the independence of eigenvectors in the relation 
tells us that $\omega^{2} \neq 1$, namely that $-1$ is not an option. Then the eigenvalue equation 
tells us that $(\overline{\omega}^{2} - 1) \overline{\omega} = \omega - \overline{\omega}$, which is to say that $\overline{\omega}^{3} = \omega$. So we simply must have that $\omega = i$, if it's an ordinary number of some sort.
posted by Kea | 10:26 AM
3 Comments:
The real Relativity, as formulated by Einstein, uses the complex i frequently, making time an imaginary quantity. Modern misinterpretations leave complex i out, and those versions have made it to textbooks. When i is used, time and space can truly be treated as the same phenomena. From this principle of causality, rather than a constant speed of light, we can derive the Lorentz transformation.
It is terrbile how your original PhD certificate was treated, but at least you are in country. If you were a terrorist risk, you would be welcomed with open arms and a welfare cheque.
Dear Kea, if you want to model "i" by real matrices, why don't you simply use
i = ((0,1),(-1,0))?
It squares to minus one, as required. The eigenvalues are i,-i, which is OK because of a Z2 automorphism, i exchanged with -i, of the complex numbers.
Lubos, the whole point of qubit MUBs is that the eigenvalues (of your sigma_Y) have to be +1,-1 and the eigenvectors have to appear in the circulant R_2. Otherwise, your idea is not a bad one.
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