Complex Conference Matrices, Complex Hadamard Matrices and Complex Equiangular Tight Frames

Et-Taoui, Boumediene

doi:10.1007/978-3-319-28186-5_16

Cited by 9 publications

(4 citation statements)

References 24 publications

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“…The frame is equal-norm if all the vectors {v j } have the same norm, and the frame is tight if the "frame bounds" A and B are equal. The ratio of the number of vectors to the dimension of the space is known as the redundancy of the frame [36]. For more on this terminology and its history, we refer to Kovačević and Chebira [37,38].…”

Section: Introductionmentioning

confidence: 99%

The SIC Question: History and State of Play

Fuchs

Hoang

Stacey

2017

Axioms

195

182

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Recent years have seen significant advances in the study of symmetric informationally complete (SIC) quantum measurements, also known as maximal sets of complex equiangular lines. Previously, the published record contained solutions up to dimension 67, and was with high confidence complete up through dimension 50. Computer calculations have now furnished solutions in all dimensions up to 151, and in several cases beyond that, as large as dimension 844. These new solutions exhibit an additional type of symmetry beyond the basic definition of a SIC, and so verify a conjecture of Zauner in many new cases. The solutions in dimensions 68 through 121 were obtained by Andrew Scott, and his catalogue of distinct solutions is, with high confidence, complete up to dimension 90. Additional results in dimensions 122 through 151 were calculated by the authors using Scott's code. We recap the history of the problem, outline how the numerical searches were done, and pose some conjectures on how the search technique could be improved. In order to facilitate communication across disciplinary boundaries, we also present a comprehensive bibliography of SIC research.

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Section: Introductionmentioning

confidence: 99%

The SIC Question: History and State of Play

Fuchs

Hoang

Stacey

2017

Axioms

195

182

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“…Of particular note, for the case l = 1, such objects are more commonly referred to as equiangular tight frames (ETFs) and are probably the most famous and well-studied class of optimally spread packings [38,9,13,17,28,31,30,33,34,35,40,43,44,45,46,51,52,53,63,64,65]. Indeed, numerous infinite families are known to exist [34] and dozens [38,9,13,17,28,31,30,33,34,35,40,43,44,45,46,51,52,53,63,64,65] -if not hundreds -of mathematicians have contributed to this study, many of whom (see [34]) are engaged in ETF research concurrently with the preparation of this document.…”

Section: Definitionmentioning

confidence: 99%

Constructions and properties of optimally spread subspace packings via symmetric and affine block designs and mutually unbiased bases

Casazza¹,

Haas²,

Stueck³

et al. 2018

Preprint

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We continue the study of optimal chordal packings, with emphasis on packing subspaces of dimension greater than one. Following a principle outlined in [12], where the authors use maximal affine block designs and maximal sets of mutually unbiased bases to construct Grassmannian 2-designs, we show that their method extends to other types of block designs, leading to a plethora of optimal subspace packings characterized by the orthoplex bound. More generally, we show that any optimal chordal packing is necessarily a fusion frame and that its spatial complement is also optimal.1991 Mathematics Subject Classification. 42C15. Key words and phrases. optimal chordal packings, tight fusion frames, orthoplex bound, mutually unbiased bases, block designs.

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“…Systems of equiangular lines -line sets with angle sets of cardinality one -are perhaps the most well-studied systems [56,29,53,28,46,9,26,11] because, according to a lower bound of Welch [58], they can form optimal packings. However, it is well-known that when the number of lines is too large relative to the dimension of the ambient vector space, then they cannot be equiangular [39,38]; furthermore, there are cases where the number does not exceed this threshold for which equiangular configurations are not possible [28,57].…”

Section: Introductionmentioning

confidence: 99%

Toward the classification of biangular harmonic frames

Casazza

Farzannia

Haas

et al. 2019

Applied and Computational Harmonic Analysis

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Equiangular tight frames (ETFs) and biangular tight frames (BTFs) -sets of unit vectors with basis-like properties whose pairwise absolute inner products admit exactly one or two values, respectivelyare useful for many applications. A well-understood class of ETFs are those which manifest as harmonic frames -vector sets defined in terms of the characters of finite abelian groups -because they are characterized by combinatorial objects called difference sets.This work is dedicated to the study of the underlying combinatorial structures of harmonic BTFs. We show that if a harmonic frame is generated by a divisible difference set, a partial difference set or by a special structure with certain Gauss summing properties -all three of which are generalizations of difference sets that fall under the umbrella term "bidifference set" -then it is either a BTF or an ETF. However, we also show that the relationship between harmonic BTFs and bidifference sets is not as straightforward as the correspondence between harmonic ETFs and difference sets, as there are examples of bidifference sets that do not generate harmonic BTFs. In addition, we study another class of combinatorial structures, the nested divisible difference sets, which yields an example of a harmonic BTF that is not generated by a bidifference set.

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Complex Conference Matrices, Complex Hadamard Matrices and Complex Equiangular Tight Frames

Cited by 9 publications

References 24 publications

The SIC Question: History and State of Play

The SIC Question: History and State of Play

Constructions and properties of optimally spread subspace packings via symmetric and affine block designs and mutually unbiased bases

Toward the classification of biangular harmonic frames

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