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arXiv:1611.09087 · Foundations of Physics

The Number Behind the Simplest SIC–POVM

Ingemar Bengtsson

Published Mar 8, 2017• 38 citations
arXivHTMLPDF

Abstract

The simple concept of a SIC poses a very deep problem in algebraic number theory, as soon as the dimension of Hilbert space exceeds three. A detailed description of the simplest possible example is given.
DOI
10.1007/s10701-017-0078-3

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A set of d 2 vectors in a Hilbert space of dimension d is called equiangular if each pair of vectors encloses the same angle. The projection operators onto these vectors define a POVM which is distinguished by its high degree of symmetry. Measures of this kind are called symmetric informationally complete , or SIC POVMs for short, and could be applied for quantum state tomography. Despite its simple geometrical description, the problem of constructing SIC POVMs or even proving their existence seems to be very hard. It is our purpose to introduce two applications of discrete Wigner functions to the analysis of the problem at hand. First, we will present a method for identifying symmetries of SIC POVMs under Clifford operations. This constitutes an alternative approach to a structure described before by Zauner and Appleby. Further, a simple and geometrically motivated construction for an SIC POVM in dimensions two and three is given (which, unfortunately, allows no generalization). Even though no new structures are found, we hope that the re-formulation of the problem may prove useful for future inquiries.
The notion of SIC-POVMs comes from quantum theory, and they were not on the horizon when I was Lars Brink’s student in the early 1980s. In the summer of 2022, I told Lars that I know how to use number theoretical insights to construct SIC-POVMs in any Hilbert space of dimension [Formula: see text], and that the construction provides a geometric setting for some deep number theoretical conjectures. I will give a sketch of this development, of what it was like to be Lars’ student, and of what his reaction to our construction was.