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

Abstract

We address the problem of constructing positive operator-valued measures (POVMs) in finite dimension n consisting of n{sup 2} 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 C{sup n} 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.

Authors:
; ; ;  [1]
  1. Department of Computer Science, Texas A and M University, College Station, Texas 77843-3112 (United States)
Publication Date:
OSTI Identifier:
20699335
Resource Type:
Journal Article
Journal Name:
Journal of Mathematical Physics
Additional Journal Information:
Journal Volume: 46; Journal Issue: 8; Other Information: DOI: 10.1063/1.1998831; (c) 2005 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; INFORMATION THEORY; MATHEMATICAL OPERATORS; MATRIX ELEMENTS; MEASURE THEORY; QUANTUM COMPUTERS; QUANTUM MECHANICS; SET THEORY; VECTORS

Citation Formats

Klappenecker, Andreas, Roetteler, Martin, Shparlinski, Igor E, Winterhof, Arne, NEC Laboratories America, Inc., Princeton, New Jersey 08540, Department of Computing, Macquarie University, Sydney, NSW 2109, and Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences Altenbergerstr. 69, 4040 Linz. On approximately symmetric informationally complete positive operator-valued measures and related systems of quantum states. United States: N. p., 2005. Web. doi:10.1063/1.1998831.
Klappenecker, Andreas, Roetteler, Martin, Shparlinski, Igor E, Winterhof, Arne, NEC Laboratories America, Inc., Princeton, New Jersey 08540, Department of Computing, Macquarie University, Sydney, NSW 2109, & Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences Altenbergerstr. 69, 4040 Linz. On approximately symmetric informationally complete positive operator-valued measures and related systems of quantum states. United States. https://doi.org/10.1063/1.1998831
Klappenecker, Andreas, Roetteler, Martin, Shparlinski, Igor E, Winterhof, Arne, NEC Laboratories America, Inc., Princeton, New Jersey 08540, Department of Computing, Macquarie University, Sydney, NSW 2109, and Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences Altenbergerstr. 69, 4040 Linz. 2005. "On approximately symmetric informationally complete positive operator-valued measures and related systems of quantum states". United States. https://doi.org/10.1063/1.1998831.
@article{osti_20699335,
title = {On approximately symmetric informationally complete positive operator-valued measures and related systems of quantum states},
author = {Klappenecker, Andreas and Roetteler, Martin and Shparlinski, Igor E and Winterhof, Arne and NEC Laboratories America, Inc., Princeton, New Jersey 08540 and Department of Computing, Macquarie University, Sydney, NSW 2109 and Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences Altenbergerstr. 69, 4040 Linz},
abstractNote = {We address the problem of constructing positive operator-valued measures (POVMs) in finite dimension n consisting of n{sup 2} 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 C{sup n} 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.},
doi = {10.1063/1.1998831},
url = {https://www.osti.gov/biblio/20699335}, journal = {Journal of Mathematical Physics},
issn = {0022-2488},
number = 8,
volume = 46,
place = {United States},
year = {Mon Aug 01 00:00:00 EDT 2005},
month = {Mon Aug 01 00:00:00 EDT 2005}
}