Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Bulletin of the Australian Mathematical Society

MORE CONSTRUCTIONS OF APPROXIMATELY MUTUALLY UNBIASED BASES

Xiwang Cao, Wun-Seng Chou

Published Aug 17, 2015• 5 citations
Publisher

Abstract

Let $m$ be a positive integer and $p$ a prime number. We prove the orthogonality of some character sums over the finite field $\mathbb{F}_{p^{m}}$ or over a subset of a finite field and use this to construct some new approximately mutually unbiased bases of dimension $p^{m}$ over the complex number field $\mathbb{C}$ , especially with $p=2$ .

DOI
10.1017/s0004972715000994

Related papers

New Constructions of Approximately Mutually Unbiased Bases by Character Sums over Galois Rings

You Gao, Ming-Yue XIE, Gang Wang +1 more

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].

IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences•Feb 6, 2024•0 citations

MUTUALLY UNBIASED BASES 1

Ingemar Bengtsson

This is a review of the problem of Mutually Unbiased Bases in finite dimensional Hilbert spaces, real and complex. Also a geometric measure of “mubness” is introduced, and applied to some explicit calculations in six di

•Jan 1, 2006•0 citations

Aspects of mutually unbiased bases in odd-prime-power dimensions

S. Chaturvedi

We rephrase the Wootters-Fields construction [W. K. Wootters and B. C. Fields, Ann. Phys. 191, 363 (1989)] of a full set of mutually unbiased bases in a complex vector space of dimensions ${N=p}^{r},$ where p is an odd prime, in terms of the character vectors of the cyclic group G of order p. This form may be useful in explicitly writing down mutually unbiased bases for ${N=p}^{r}.$

arXiv:quant-ph/0109003•Physical Review A•Mar 21, 2002•59 citations

Two new constructions of approximately mutually unbiased bases

Gang Wang, Min-Yao Niu, Fang‐Wei Fu

Two orthonormal bases [Formula: see text] and [Formula: see text] of a [Formula: see text]-dimensional complex inner-product space [Formula: see text] are called mutually unbiased bases (MUBs) if and only if [Formula: see text] holds for all [Formula: see text] and [Formula: see text]. The size of any set containing pairwise MUBs of [Formula: see text] cannot exceed [Formula: see text]. If [Formula: see text] is a power of a prime, then extremal sets containing [Formula: see text] MUBs are known to exist, which are called the complete MUBs of [Formula: see text]. We have not known whether there exist complete MUBs when [Formula: see text] is not a power of a prime so far. Therefore, many researchers focus their attention on approximately mutually unbiased bases (AMUB). In this paper, two new constructions of AMUB of [Formula: see text] are provided based on the mixed character sum of two special kinds of functions over finite fields.

International Journal of Quantum Information•Jun 1, 2018•4 citations

The Mutually Unbiased Bases Revisited

Monique Combescure

The study of Mutually Unbiased Bases continues to be developed vigorously, and presents several challenges in the Quantum Information Theory. Two orthonormal bases in $\mathbb C^d, B {and} B'$ are said mutually unbiased if $\forall b\in B, b'\in B'$ the scalar product $b\cdot b'$ has modulus $d^{-1/2}$. In particular this property has been introduced in order to allow an optimization of the measurement-driven quantum evolution process of any state $ψ\in \mathbb C^d$ when measured in the mutually unbiased bases $B\_{j} {of} \mathbb C^d$. At present it is an open problem to find the maximal umber of mutually Unbiased Bases when $d$ is not a power of a prime number. \noindent In this article, we revisit the problem of finding Mutually Unbiased Bases (MUB's) in any dimension $d$. The method is very elementary, using the simple unitary matrices introduced by Schwinger in 1960, together with their diagonalizations. The Vandermonde matrix based on the $d$-th roots of unity plays a major role. This allows us to show the existence of a set of 3 MUB's in any dimension, to give conditions for existence of more than 3 MUB's for $d$ even or odd number, and to recover the known result of existence of $d+1$ MUB's for $d$ a prime number. Furthermore the construction of these MUB's is very explicit. As a by-product, we recover results about Gauss Sums, known in number theory, but which have apparently not been previously derived from MUB properties.

arXiv:quant-ph/0605090•May 10, 2006•4 citations

A SU (2) RECIPE FOR MUTUALLY UNBIASED BASES

M. Kibler, Michel R. P. Planat

A simple recipe for generating a complete set of mutually unbiased bases in dimension 2j + 1, with 2j integer and 2j + 1 prime, is developed from a single matrix Va acting on a space of constant angular momentum j and defined in terms of the irreducible characters of the cyclic group C2j + 1. This recipe yields an (apparently new) compact formula for the vectors spanning the various mutually unbiased bases. In dimension (2j + 1)e, with 2j integer, 2j + 1 prime and e positive integer, the use of direct products of matrices of type Va makes it possible to generate mutually unbiased bases. As two pending results, the matrix Va is used in the derivation of a polar decomposition of SU (2) and of a FFZ algebra.

arXiv:quant-ph/0601092•International Journal of Modern Physics B•May 20, 2006•79 citations
Let mmm be a positive integer and ppp a prime number. We prove the orthogonality of some character sums over the finite field Fpm\mathbb{F}_{p^{m}}Fpm​ or over a subset of a finite field and use this to construct some new approximately mutually unbiased bases of dimension pmp^{m}pm over the complex number field C\mathbb{C}C , especially with p=2p=2p=2 .
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].
We rephrase the Wootters-Fields construction [W. K. Wootters and B. C. Fields, Ann. Phys. 191, 363 (1989)] of a full set of mutually unbiased bases in a complex vector space of dimensions N=pr,{N=p}^{r},N=pr, where p is an odd prime, in terms of the character vectors of the cyclic group G of order p. This form may be useful in explicitly writing down mutually unbiased bases for N=pr.{N=p}^{r}.N=pr.
Two orthonormal bases [Formula: see text] and [Formula: see text] of a [Formula: see text]-dimensional complex inner-product space [Formula: see text] are called mutually unbiased bases (MUBs) if and only if [Formula: see text] holds for all [Formula: see text] and [Formula: see text]. The size of any set containing pairwise MUBs of [Formula: see text] cannot exceed [Formula: see text]. If [Formula: see text] is a power of a prime, then extremal sets containing [Formula: see text] MUBs are known to exist, which are called the complete MUBs of [Formula: see text]. We have not known whether there exist complete MUBs when [Formula: see text] is not a power of a prime so far. Therefore, many researchers focus their attention on approximately mutually unbiased bases (AMUB). In this paper, two new constructions of AMUB of [Formula: see text] are provided based on the mixed character sum of two special kinds of functions over finite fields.
The study of Mutually Unbiased Bases continues to be developed vigorously, and presents several challenges in the Quantum Information Theory. Two orthonormal bases in Cd,BandB′\mathbb C^d, B {and} B'Cd,BandB′ are said mutually unbiased if ∀b∈B,b′∈B′\forall b\in B, b'\in B'∀b∈B,b′∈B′ the scalar product b⋅b′b\cdot b'b⋅b′ has modulus d−1/2d^{-1/2}d−1/2. In particular this property has been introduced in order to allow an optimization of the measurement-driven quantum evolution process of any state ψ∈Cdψ\in \mathbb C^dψ∈Cd when measured in the mutually unbiased bases B_jofCdB\_{j} {of} \mathbb C^dB_jofCd. At present it is an open problem to find the maximal umber of mutually Unbiased Bases when ddd is not a power of a prime number. \noindent In this article, we revisit the problem of finding Mutually Unbiased Bases (MUB's) in any dimension ddd. The method is very elementary, using the simple unitary matrices introduced by Schwinger in 1960, together with their diagonalizations. The Vandermonde matrix based on the ddd-th roots of unity plays a major role. This allows us to show the existence of a set of 3 MUB's in any dimension, to give conditions for existence of more than 3 MUB's for ddd even or odd number, and to recover the known result of existence of d+1d+1d+1 MUB's for ddd a prime number. Furthermore the construction of these MUB's is very explicit. As a by-product, we recover results about Gauss Sums, known in number theory, but which have apparently not been previously derived from MUB properties.
A simple recipe for generating a complete set of mutually unbiased bases in dimension 2j + 1, with 2j integer and 2j + 1 prime, is developed from a single matrix Va acting on a space of constant angular momentum j and defined in terms of the irreducible characters of the cyclic group C2j + 1. This recipe yields an (apparently new) compact formula for the vectors spanning the various mutually unbiased bases. In dimension (2j + 1)e, with 2j integer, 2j + 1 prime and e positive integer, the use of direct products of matrices of type Va makes it possible to generate mutually unbiased bases. As two pending results, the matrix Va is used in the derivation of a polar decomposition of SU (2) and of a FFZ algebra.