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arXiv · 1612.02351

Phase-space representations of SIC-POVM fiducial states

Marcos Saraceno·Leonardo Ermann·Cecilia Cormick
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Abstract

The problem of finding symmetric informationally complete POVMs (SIC-POVMs) has been solved numerically for all dimensions $d$ up to 67 (A.J. Scott and M. Grassl, {\it J. Math. Phys.} 51:042203, 2010), but a general proof of existence is still lacking. For each dimension, it was shown that it is possible to find a SIC-POVM which is generated from a fiducial state upon application of the operators of the Heisenberg-Weyl group. We draw on the numerically determined fiducial states to study their phase-space features, as displayed by the characteristic function and the Wigner, Bargmann and Husimi representations, adapted to a Hilbert space of finite dimension. We analyze the phase-space localization of fiducial states, and observe that the SIC-POVM condition is equivalent to a maximal delocalization property. Finally, we explore the consequences in phase space of the conjectured Zauner symmetry. In particular, we construct an Hermitian operator commuting with this symmetry that leads to a representation of fiducial states in terms of eigenfunctions with definite semiclassical features.

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Marcos Saraceno, Leonardo Ermann, Cecilia Cormick. 2016-12-07. Phase-space representations of SIC-POVM fiducial states. https://doi.org/10.1103/physreva.95.032102

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