About SIC POVMs and discrete Wigner distributions

Samuel Colin*, John Corbett, Thomas Durt, David Gross

*Corresponding author for this work

    Research output: Contribution to journalArticle

    30 Citations (Scopus)

    Abstract

    A set of d2 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.

    Original languageEnglish
    Pages (from-to)S778-S785
    Number of pages8
    JournalJournal of Optics B: Quantum and Semiclassical Optics
    Volume7
    Issue number12
    DOIs
    Publication statusPublished - 1 Dec 2005

    Keywords

    • Clifford group
    • Discrete Wigner distributions
    • Equiangular lines
    • Optimal quantum tomography
    • SIC POVMs

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