In order to describe the right setting to handle Zauner's conjecture on mutually unbiased bases (MUBs) (saying that in $\mathbb{C}^d$ , a set of MUBs of the theoretical maximal size $d + 1$ exists only if $d$ is a prime power), we pose some fundamental questions which naturally arise. Some of these questions have important consequences for the construction theory of (new) sets of maximal MUBs.In order to describe the right setting to handle Zauner's conjecture on mutually unbiased bases (MUBs) (saying that in $\mathbb{C}^d$, a set of MUBs of the theoretical maximal size $d + 1$ exists only if $d$ is a prime power), we pose some fundamental questions which naturally arise. Some of these questions have important consequences for the construction theory of (new) sets of maximal MUBs.

arXiv:1409.3386Sep 11, 20140 citations

A New Proof for the Existence of Mutually Unbiased Bases

Not Available Not Available, Not Available Not Available

arXiv:quant-ph/0103162AlgorithmicaNov 6, 2002569 citations

Constructions of Approximately Mutually Unbiased Bases

Igor E. Shparlinski, Arne Winterhof

Lecture notes in computer scienceJan 1, 200625 citations

Mutually Unbiased Bases in Composite Dimensions – A Review

Daniel McNulty, Stefan Weigert

Maximal sets of mutually unbiased bases are useful throughout quantum physics, both in a foundational context and for applications. To date, it remains unknown if complete sets of mutually unbiased bases exist in Hilbert spaces of dimensions different from a prime power, i.e. in composite dimensions such as six or ten. Fourteen mathematically equivalent formulations of the existence problem are presented. We comprehensively summarise analytic, computer-aided and numerical results relevant to the case of composite dimensions. Known modifications of the existence problem are reviewed and potential solution strategies are outlined.Maximal sets of mutually unbiased bases are useful throughout quantum physics, both in a foundational context and for applications. To date, it remains unknown if complete sets of mutually unbiased bases exist in Hilbert spaces of dimensions different from a prime power, i.e. in composite dimensions such as six or ten. Fourteen mathematically equivalent formulations of the existence problem are presented. We comprehensively summarise analytic, computer-aided and numerical results relevant to the case of composite dimensions. Known modifications of the existence problem are reviewed and potential solution strategies are outlined.

arXiv:2410.23997QuantumApr 1, 20268 citations

Constructions of Mutually Unbiased Bases

Andreas Klappenecker, Martin Rötteler

Lecture notes in computer scienceJan 1, 2004288 citations

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].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 SciencesFeb 6, 20240 citations

In order to describe the right setting to handle Zauner's conjecture on mutually unbiased bases (MUBs) (saying that in

\mathbb{C}^d

, a set of MUBs of the theoretical maximal size

d + 1

exists only if

d

is a prime power), we pose some fundamental questions which naturally arise. Some of these questions have important consequences for the construction theory of (new) sets of maximal MUBs.