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.

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Planat scited SICs and Algebraic Number Theory

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Planat commented on The magic of universal quantum computing with permutations

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.

Mar 02 2014 07:47 UTC

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Dec 15 2012 08:45 UTC

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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.