On approximately symmetric informationally complete positive operator-valued measures and related systems of quantum states

Andreas Klappenecker, Martin Rötteler, Igor E. Shparlinski, Arne Winterhof

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Abstract

We address the problem of constructing positive operator-valued measures (POVMs) in finite dimension n consisting of n2 operators of rank one which have an inner product close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. However, SIC-POVMs are notoriously hard to construct and, despite some success of constructing them numerically, there is no analytic construction known. We present two constructions of approximate versions of SIC-POVMs, where a small deviation from uniformity of the inner products is allowed. The first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power. The second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases. Moreover, we construct vector systems in Cn which are almost orthogonal and which might turn out to be useful for quantum computation. Our constructions are based on results of analytic number theory.

Original languageEnglish
Article number082104
Pages (from-to)1-17
Number of pages17
JournalJournal of Mathematical Physics
Volume46
Issue number8
DOIs
Publication statusPublished - Aug 2005

Bibliographical note

Copyright 2005 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in Journal of mathematical physics, Vol. 46, Issue 8, pp.082104 (17pgs), and may be found at http://link.aip.org/link/?jmp/46/082104.

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