On Weyl-Heisenberg orbits of equiangular lines

Khatirinejad, Mahdad

doi:10.1007/s10801-007-0104-1

Cited by 32 publications

(28 citation statements)

References 11 publications

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“…We note that Theorem 1 has recently been independently discovered by other researchers [2,11]. As explained in the next section, Theorems 1 and 2 imply any search for GMEF generators must necessarily consider the role of chirps, that is, unimodular functions of nonconstant frequency.…”

Section: Theorem 2 For Anymentioning

confidence: 77%

Maximally Equiangular Frames and Gauss Sums

Fickus

2009

J Fourier Anal Appl

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In a finite-dimensional complex Euclidean space, a maximally equiangular frame is a tight frame which has a number of elements equal to the square of the dimension of the space, and in which the inner products of distinct elements are of constant magnitude. Though the general question of their existence remains open, many examples of maximally equiangular frames have been constructed as finite Gabor systems. These constructions involve number theory, specifically Schaar's identity, which provides a reciprocity formula for quadratic Gauss sums. To be precise, Zauner used Schaar's identity to compute the spectrum of a chirp-Fourier operator, the eigenvectors of which he conjectured to be well-suited for the construction of maximally equiangular Gabor frames. We provide two new characterizations of such frames, both of which further confirm the relevance of the theory of Gauss sums to this area of frame theory. We also show how the unique time-frequency properties of a particular cyclic chirp function may be exploited to provide a new, short and elementary proof of Schaar's identity.

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Section: Theorem 2 For Anymentioning

confidence: 77%

Maximally Equiangular Frames and Gauss Sums

Fickus

2009

J Fourier Anal Appl

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“…Using Thus the conclusion, when the dimension is a prime equal to 1 modulo 3, is that the 2(p − 1) Alltop vectors in a given Zauner subspace are to be found in equal numbers in its intersections with two real subspaces. For p = 7, 19 there also exist SIC vectors in these intersections [19], but this does not seem to happen for any other value of p [35].…”

Section: Further Symmetries Of Alltop Vectorsmentioning

confidence: 99%

Order 3 symmetry in the Clifford hierarchy

Bengtsson¹,

Blanchfield²,

Campbell³

et al. 2014

J. Phys. A: Math. Theor.

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We investigate the action of the first three levels of the Clifford hierarchy on sets of mutually unbiased bases comprising the Ivanovic MUB and the Alltop MUBs. Vectors in the Alltop MUBs exhibit additional symmetries when the dimension is a prime number equal to 1 modulo 3 and thus the set of all Alltop vectors splits into three Clifford orbits. These vectors form configurations with so-called Zauner subspaces, eigenspaces of order 3 elements of the Clifford group highly relevant to the SIC problem. We identify Alltop vectors as the magic states that appear in the context of fault-tolerant universal quantum computing, wherein the appearance of distinct Clifford orbits implies a surprising inequivalence between some magic states.

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“…Equiangular lines have been studied for over 65 years [13], and their construction remains "[o]ne of the most challenging problems in algebraic combinatorics" [16]. In particular, the study of equiangular lines in complex space has intensified recently, as its importance in quantum information theory has become apparent [1,9,17,18].…”

Section: Introductionmentioning

confidence: 99%

A Simple Construction of Complex Equiangular Lines

Jedwab

Wiebe

2015

Springer Proceedings in Mathematics &Amp; Statistics

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Abstract. A set of vectors of equal norm in C d represents equiangular lines if the magnitudes of the inner product of every pair of distinct vectors in the set are equal. The maximum size of such a set is d 2 , and it is conjectured that sets of this maximum size exist in C d for every d ≥ 2. We describe a new construction for maximum-sized sets of equiangular lines, exposing a previously unrecognized connection with Hadamard matrices. The construction produces a maximum-sized set of equiangular lines in dimensions 2, 3 and 8.

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On Weyl-Heisenberg orbits of equiangular lines

Cited by 32 publications

References 11 publications

Maximally Equiangular Frames and Gauss Sums

Maximally Equiangular Frames and Gauss Sums

Order 3 symmetry in the Clifford hierarchy

A Simple Construction of Complex Equiangular Lines

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