Okay, I see the resolution to my confusion now (and admit that I was confused). Thanks to Michel, Marcus, Blake, and Steve!
Since I don't know the first thing about cyclotomic field norms... can anybody explain the utility of this norm, for this problem? I mean, just to be extreme, I could define a trivial norm that is 1 for all vectors except $\vec{0}$, and then all rank-1 POVMs would be equiangular. I'm not by any means saying that this is what's done here! My point is that there exist norms for which equi-angularity is less interesting than others. The Hilbert-Schmidt norm is very relevant for quantum states in Hilbert spaces, because it's what appears in Born's rule. What can I do with this field norm that makes it interesting and relevant?
P.S. @Marcus, if I'm understanding this correctly, then whenever two pairs have equal Hilbert-Schmidt norm, they will have equal field norm (but different H-S norms can correspond to equal field norms). So SICs should still be equiangular in field norm. Unless I'm misunderstanding again!
Okay, I see the resolution to my confusion now (and admit that I was confused). Thanks to Michel, Marcus, Blake, and Steve!
Since I don't know the first thing about cyclotomic field norms... can anybody explain the utility of this norm, for this problem? I mean, just to be extreme, I could define a trivial norm that is 1 for all vectors except $\vec{0}$, and then all rank-1 POVMs would be equiangular. I'm not by any means saying that this is what's done here! My point is that there exist norms for which equi-angularity is less interesting than others. The Hilbert-Schmidt norm is very relevant for quantum states in Hilbert spaces, because it's what appears in Born's rule. What can I do with this field norm that makes it interesting and relevant?
Okay, I see the resolution to my confusion now (and admit that I was confused).
Since I don't know the first thing about cyclotomic field norms... can anybody explain the utility of this norm, for this problem? I mean, just to be extreme, I could define a trivial norm that is 1 for all vectors except $\vec{0}$, and then all rank-1 POVMs would be equiangular. I'm not by any means saying that this is what's done here! My point is that there exist norms for which equi-angularity is less interesting than others. The Hilbert-Schmidt norm is very relevant for quantum states in Hilbert spaces, because it's what appears in Born's rule. What can I do with this field norm that makes it interesting and relevant?