Planat

Planat148

Jun 14 2017 11:54 UTC
May 11 2017 09:03 UTC

Dear Christopher,

1. Could you comment on the connection to the fine structure constant in footnote 15 in which you write "Implicit in it is the number 137!"?

2. Would the Qbism philosophy be destroyed by restricting to IC's instead of SICs as in https://scirate.com/arxiv/1704.02749#807?

Thanks.

May 11 2017 08:56 UTC
Apr 14 2017 08:11 UTC

First of all, thanks to all for helping to clarify some hidden points of our paper.
As you can see, the field norm generalizes the standard Hilbert-Schmidt norm.
It works for SIC [e.g. d=2, d=3 (the Hesse) and d=8 (the Hoggar)].

The first non-trivial case is with d=4 when one needs to extend the rational field
by a 12th root of unity, i.e. n=GCD(d,r)=GCD(4,3)=12, that is r=3 for defining the appropriate
fiducial state and d=4 to allow the action of the two-qubit Pauli group on it.
Then one needs the field norm in the so defined cyclotomic extension to normalize the vectors of the
resulting IC-POVM. This IC is dichotomic in angles and traces of paiwise products.

Incidently, such a 4-dimensional IC is related to the Mermin square through the traces of triple products.

Apr 13 2017 16:49 UTC

To define the complex angle, we used the (cyclotomic) field norm to the power one over the degree of the field, as stated in the introduction. It recovers the particular case of angles for SICs. In this sense "equiangular" means that all pairs of distinct lines make the same angle.

Apr 13 2017 07:14 UTC

The trace of pairwise product of (distinct) projectors is not constant. For example, with the state $(0,1,-1,-1,1)$, one gets an equiangular IC-POVM in which the trace is trivalued: it is either $1/16$, or $(7 \pm 3\sqrt{5})/32$. For the state (0,1,i,-i,-1), there are five values of the trace.
We should explicit this observation in the next version of the paper.

Apr 12 2017 13:58 UTC

Yes, the IC-POVMs under consideration are minimal. The IC-POVM in dimension 5 is equiangular but is also not a SIC. In particular the trace product relation of a SIC is not satisfied. For the equiangular IC-POVM in dimension 7, we have a similar result.

Jan 30 2017 08:14 UTC
Jan 24 2017 13:21 UTC
Jan 24 2017 13:09 UTC

Are you sure? Since we do not propose a conjecture, there is nothing wrong. A class of strange states underlie the pentagons in question. The motivation is to put the magic of computation in the permutation frame, one needs more work to check its relevance.