TY - JOUR

T1 - On approximately symmetric informationally complete positive operator-valued measures and related systems of quantum states

AU - Klappenecker, Andreas

AU - Rötteler, Martin

AU - Shparlinski, Igor E.

AU - Winterhof, Arne

N1 - 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.

PY - 2005/8

Y1 - 2005/8

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=24344507240&partnerID=8YFLogxK

U2 - 10.1063/1.1998831

DO - 10.1063/1.1998831

M3 - Article

VL - 46

SP - 1

EP - 17

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 8

M1 - 082104

ER -