### M Theory Lesson 271

Having committed ourselves to finite field arithmetic for now, we remember that MUB operators use only a small set of truth values. The qubit case requires the field with five elements, whereas qutrits only require the four element field.

But the factor of $i$ required in the qubit case means that one does not automatically encounter (at least in quantum mechanics) the mod $7$ multi muon arithmetic, because six dimensions uses $12$th roots of unity. However, the main obstacle to using square roots in the qubit case, namely the smallness of the number $2$, no longer applies in dimension $6$. Thus there should be a measurement operator set based on the seven element field for the important dimension six case.

The interesting case from the point of view of modular mathematics is the number $24$, which initially appears for (stringy $F$ theory) dimension $12$, combining mass and spin quantum numbers. Recall that these dimensions are also counted by the triple of Riemann moduli spaces, $M_{0,6}$, $M_{1,3}$ and $M_{2,0}$, each of twistor space dimension.

But the factor of $i$ required in the qubit case means that one does not automatically encounter (at least in quantum mechanics) the mod $7$ multi muon arithmetic, because six dimensions uses $12$th roots of unity. However, the main obstacle to using square roots in the qubit case, namely the smallness of the number $2$, no longer applies in dimension $6$. Thus there should be a measurement operator set based on the seven element field for the important dimension six case.

The interesting case from the point of view of modular mathematics is the number $24$, which initially appears for (stringy $F$ theory) dimension $12$, combining mass and spin quantum numbers. Recall that these dimensions are also counted by the triple of Riemann moduli spaces, $M_{0,6}$, $M_{1,3}$ and $M_{2,0}$, each of twistor space dimension.

## 1 Comments:

Kea, More ideas on proving that every unitary matrix can be written in complex phase times magic form.The idea is that row and column multiply by phases defines an equivalence class on matrices and unitary matrices. Magic matrices form a subalgebra of all matrices and the magic unitary matrices are a subgroup of unitary matrices, which are a subgroup of all matrices. As you've noted before, magic matrices are equivalent, under Fourier transform, to 1x1+2x2 block diagonal matrices. The same applies to unitary magic matrices.

The Fourier converted question is can you convert a unitary matrix to block diagonal form with magic matrices, i.e.

U to B = MUN

where M and N are magic unitary.

All the arrows make it seem like it should be in category theory, hence my putting this comment here.

Getting back to the subject of this post, I'm wondering if it is useful to consider finite subgroups.

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