Blake Stacey in reply to Steve Flammia Apr 14 2017 00:35 UTC

I agree with Steve Flammia's comment. The field norm is a nice generalization of the standard norm. (I haven't yet thought about whether there might be a physics motivation for it, rather than a purely mathematical one, but that's not important right now.) To avoid confusion, some phrase like "equiangular with respect to the field norm" or "field-norm equiangular" should be used.

Blake Stacey in reply to Planat Apr 12 2017 18:20 UTC

This is why I am confused (it is probably just a reading comprehension error on my part): If the POVM is IC, it must have at least $d^2$ elements. If it is a minimal IC-POVM, it must have exactly $d^2$ elements. But if it is minimal, IC and equiangular, then the angle is fixed by the requirement that the elements sum to the identity. Suppose that the trace of $\Pi_i \Pi_j$ is $\alpha$ whenever $i \neq j$. Summing this over all $j$ yields $1 + (d^2-1)\alpha$. But the projectors $\Pi_i$ themselves must sum to $dI$, so the value of $\alpha$ is fixed to $1/(d+1)$.

Blake Stacey Apr 12 2017 00:34 UTC

Clarification request: Are all the IC-POVMs in this paper minimal? That is, does the number of elements in each POVM equal the square of the dimension? If so, I am confused about the quoted value of the inner product between projectors for the equiangular IC-POVM in dimension 5.

Eight hundred forty-four!