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.

Eq. (14) defines the sum negativity as $\sum_u |W_u| - 1$, but there should be an overall factor of $1/2$ (see arXiv:1307.7171, definition 10). For both the Strange states and the Norrell states, the sum negativity should be $1/3$: The Strange states (a.k.a., Hesse SIC vectors) have one negative entry in their Wigner representation, while the Norrell states each have two negative entries of value $-1/6$. This makes the greater robustness of the Strange states under incoherent noise easy to see, because mixing in the garbage state hits the Norrell states twice as hard.

Edited Jan 17 2018 20:13 UTC by Blake Stacey