Dec 14 2017 07:44 UTC

Planat scited General Quantum Theory

Nov 17 2017 08:27 UTC

Nov 09 2017 07:00 UTC

Planat scited polyDB: A Database for Polytopes and Related Objects

Nov 03 2017 07:37 UTC

Oct 23 2017 07:36 UTC

Sep 20 2017 07:38 UTC

Sep 11 2017 07:08 UTC

Jun 14 2017 11:54 UTC

Planat scited Preprint Déjà Vu: an FAQ

May 11 2017 09:03 UTC

Planat commented on Notwithstanding Bohr, the Reasons for QBism

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

Planat scited Notwithstanding Bohr, the Reasons for QBism

Apr 14 2017 08:11 UTC

Planat commented on Magic informationally complete POVMs with permutations

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

Planat commented on Magic informationally complete POVMs with permutations

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

Planat commented on Magic informationally complete POVMs with permutations

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

Planat commented on Magic informationally complete POVMs with permutations

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.

Apr 07 2017 06:45 UTC

Apr 06 2017 08:48 UTC

Mar 20 2017 08:17 UTC

Feb 24 2017 09:01 UTC

Jan 30 2017 08:14 UTC

Planat scited On discrete structures in finite Hilbert spaces

Jan 27 2017 07:45 UTC

Jan 26 2017 07:52 UTC

Jan 24 2017 13:21 UTC

Planat scited SICs and Algebraic Number Theory

Jan 24 2017 13:19 UTC

Jan 24 2017 13:09 UTC

Interesting work. You don't require that the polar space has to be symplectic. In ordinary quantum mechanics the commutation of n-qudit observables is ruled by a symplectic polar space. For two qubits, it is the generalized quadrangle GQ(2,2). Incidently, in https://arxiv.org/abs/1601.04865 this problem is related to general simple groups.