arXiv subjects

Helena Granstrom

Publications and source records attributed to Helena Granstrom.

3 recordsLinked to original sources

arXiv2009

Pentagrams and paradoxes

Klyachko and coworkers consider an orthogonality graph in the form of a pentagram, and in this way derive a Kochen-Specker inequality for spin 1 systems. In some low-dimensional situations Hilbert spaces are naturally organised, by a magical choice of basis, into SO(N) orbits. Combining these ideas some very elegant results emerge. We give a careful discussion of the pentagram operator, and then show how the pentagram underlies a number of other quantum "paradoxes", such as that of Hardy.

quant-ph↗

arXiv2008

The Frame Potential, on Average

A SIC consists of N^2 equiangular unit vectors in an N dimensional Hilbert space. The frame potential is a function of N^2 unit vectors. It has a unique global minimum if the vectors form a SIC, and this property has been made use of in numerical searches for SICs. When the vectors form an orbit of the Heisenberg group the frame potential becomes a function of a single fiducial vector. We analytically compute the average of this function over Hilbert space. We also compute averages when the fiducial vector is placed in certain special subspaces defined by the Clifford group.

quant-ph↗

arXiv2007

Some remarks on the theorems of Gleason and Kochen-Specker

A Gleason-type theorem is proved for two restricted classes of informationally complete POVMs in the qubit case. A particular (incomplete) Kochen-Specker colouring, suggested by Appleby in dimension three, is generalized to arbitrary dimension. We investigate its effectivity as a function of dimension, using two different measures of this. In particular, we will derive a limit for the fraction of the sphere that can be satisfactorily coloured using the generalized Appleby construction as the number of dimensions approaches infinity. The second, and physically more relevant measure of effectivity, is to look at the fraction of possible ON-bases properly coloured. Using this measure, we will derive a 'lower bound for the upper bound' in three and four real dimensions.

quant-ph↗