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Scientia Sinica Mathematica

利用Gauss 和与Jacobi 和构造近似MUB 和SIC-POVM

AiXian ZHANG, WeiYang WANG, Keqin Feng

Published Oct 1, 2012• 9 citations
Publisher

Abstract

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 SIC- POVMs in Ck 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 Ck 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.

DOI
10.1360/012012-186

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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 SIC- POVMs in Ck 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 Ck 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.
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.
A symmetric informationally complete positive operator-valued measure (SIC-POVM) is a POVM in [Formula: see text] consisting of [Formula: see text] positive operators of rank one such that all of whose Hermite inner products are equal. SIC-POVMs are important in quantum information theory, which have many applications in quantum state tomography, quantum cryptography and basic research in quantum mechanics. However, it is very difficult to construct SIC-POVMs. Therefore, many scholars have focused on approximately symmetric informationally complete positive operator-valued measures (ASIC-POVMs) for which the Hermite inner products are close to equal. In this paper, two new constructions of ASIC-POVMs are provided by using character sums and some special functions over finite fields.
The paper concerns Weyl-Heisenberg covariant SIC-POVMs (symmetric informationally complete positive operator valued measures) and full sets of MUBs (mutually unbiased bases) in prime dimension. When represented as vectors in generalized Bloch space a SIC-POVM forms a d 2 − 1 dimensional regular simplex (d being the Hilbert space dimension). By contrast, the generalized Bloch vectors representing a full set of MUBs form d + 1 mutually orthogonal d − 1 dimensional regular simplices. In this paper we show that, in the Weyl-Heisenberg case, there are some simple geometrical relationships between the single SIC-POVM simplex and the d + 1 MUB simplices. We go on to give geometrical interpretations of the minimum uncertainty states introduced by Wootters and Sussman, and by Appleby, Dang and Fuchs, and of the fiduciality condition given by Appleby, Dang and Fuchs. 1
Mutually Unbiased bases has various application in quantum information procession and coding theory. There can be maximum d + 1 MUBs in C^d and d/2 +1 MUBs in R^d. But , over R^d MUBs are known to be non existent when d is odd and for most of the other even d there are mostly 3 Real MUBs. In case of C^d the construction for complete set of MUBs are known for only Prime Power dimension. Thus in general large set of MUBs are not known, particularly for composite dimensions which are not of the form of prime powers. Because of this, there are many constructions of Approximate version of MUBs. In this paper we make an attempt to define certain measures to characterise the AMUBs. Our construction of measures derives its inspiration from the applications of MUBs, and based on them, we define certain quantifiable measures, which are can be computed and gives estimates of how close the Approximate MUBs are to the MUBs. We use geometric interpretation, projective design features of MUBs and applications like Optimal State determination and Entropic Uncertainty of MUBs. We show generic relationship between these measures and show that it can be evaluated for APMUBs without known the exact construction details, thereby showing that definition of APMUB is sufficient completely characterise it. We also evaluate these measure for an interesting class of AMUBs called Weak MUBs and certain AMUBs constructed using RBDs.
Mutually unbiased bases (MUBs) are widely used in quantum information processing and play an important role in quantum cryptography, quantum state tomography and communications. It's difficult to construct MUBs and remains unknown whether complete MUBs exist for any non prime power. Therefore, researchers have proposed the solution to construct approximately mutually unbiased bases (AMUBs) by weakening the inner product conditions. This paper constructs q AMUBs of ℂq, (q+1) AMUBs of ℂq-1 and q AMUBs of ℂq-1 by using character sums over Galois rings and finite fields, where q is a power of a prime. The first construction of q AMUBs of ℂq is new which illustrates K AMUBs of ℂK can be achieved. The second and third constructions in this paper include the partial results about AMUBs constructed by W. Wang et al. in [9].
The paper concerns Weyl-Heisenberg covariant SIC-POVMs (symmetric informationally complete positive operator valued measures) and full sets of MUBs (mutually unbiased bases) in prime dimension. When represented as vectors in generalized Bloch space a SIC-POVM forms a d^2-1 dimensional regular simplex (d being the Hilbert space dimension). By contrast, the generalized Bloch vectors representing a full set of MUBs form d+1 mutually orthogonal d-1 dimensional regular simplices. In this paper we show that, in the Weyl-Heisenberg case, there are some simple geometrical relationships between the single SIC-POVM simplex and the d+1 MUB simplices. We go on to give geometrical interpretations of the minimum uncertainty states introduced by Wootters and Sussman, and by Appleby, Dang and Fuchs, and of the fiduciality condition given by Appleby, Dang and Fuchs.