利用Gauss和与Jacobi和构造近似MUB和SIC-POVM

Constructions of approximately mutually unbiased bases and symmetric informationally complete positive operator-valued measures by Gauss and Jacobi sums

MUB(mutually unbiased bases)和SIC-POVM(symmetric informationally completepositive operator-valued measure)是量子信息中的两个重要研究对象.目前关于非素数幂维的完全MUB是否存在还没有确定的结果,对于SIC-POVM目前只有有限多种维数K有存在性结果或数值结果.于是很多弱化了内积条件的近似MUB和SIC-POVM被人们所考虑.本文使用Klappenecker等人给出的近似MUB和SIC-POVM的定义,利用Gauss和与Jacobi和对于素数方幂q给出了一类q1维q-近似MUB(AMUB)、一类q1维(q+1)AMUB以及q+1维qAMUB,还利用Gauss和给出了一类q1维近似SIC-POVM(ASIC-POVM).

Mutually unbiased bases （MUB） and symmetric informationally complete positive operator-valued measure （SIC-POVM） are both important objects in quantum information theory.While people do not know if there exists a complete MUB for non-prime-power dimension,several versions of approximately MUB have been considered by relaxed the inner product condition.So far there are only finite number of K such that SICPOVMs in C k have been found.As in the MUB case,several versions of approximately SIC-POVM have been considered by relaxed the inner product condition.In this paper,we use the definitions of approximate MUB and SIC-POVM given by Klappenecker et al.For prime power q,we present simple constructions of q approximately MUB （AMUB） for dimension q-1,q＋1 AMUB for dimension q-1,which shows the number of orthonormal bases of an AMUB in C k can be more than K＋1,and q AMUB for dimension q＋1 by Gauss and Jacobi sums.We also present a construction of approximately SIC-POVM （ASIC-POVM） in dimension q-1 by Gauss sum.