Equiangular Tight Frames From Hyperovals

Fickus, Matthew; Mixon, Dustin G.; Jasper, John

doi:10.1109/tit.2016.2587865

Cited by 63 publications

(57 citation statements)

References 37 publications

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“…This results in a collection of vectors in

C^{false| m false|}

which look like the vectors in

normalΦ

padded with

| m | - | D |

zero rows. Thus, the vectors form an equiangular tight frame for their span . It was further shown in that equiangular tight frames formed from the so‐called McFarland difference sets are equivalent to certain Steiner equiangular tight frames.…”

Section: Gabor–steiner Equiangular Tight Framesmentioning

confidence: 98%

Optimal arrangements of classical and quantum states with limited purity

Bodmann

King

2019

J. London Math. Soc.

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We consider sets of trace‐normalized non‐negative operators in Hilbert–Schmidt balls that maximize their mutual Hilbert–Schmidt distance; these are optimal arrangements in the sets of purity‐limited classical or quantum states on a finite‐dimensional Hilbert space. Classical states are understood to be represented by diagonal matrices, with the diagonal entries forming a probability vector. We also introduce the concept of spectrahedron arrangements which provides a unified framework for classical and quantum arrangements and the flexibility to define new types of optimal packings. Continuing a prior work, we combine combinatorial structures and line packings associated with frames to arrive at optimal arrangements of higher rank quantum states. One new construction that is presented involves generating an optimal arrangement we call a Gabor–Steiner equiangular tight frame as the orbit of a projective representation of the Weyl–Heisenberg group over any finite abelian group. The minimal sets of linearly dependent vectors, the so‐called binder, of the Gabor–Steiner equiangular tight frames are then characterized; under certain conditions, these form combinatorial block designs and in one case generate a new class of block designs. The projections onto the span of minimal linearly dependent sets in the Gabor–Steiner equiangular tight frame are then used to generate further optimal spectrahedron arrangements.

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“…This results in a collection of vectors in

C^{false| m false|}

which look like the vectors in

normalΦ

padded with

| m | - | D |

Section: Gabor–steiner Equiangular Tight Framesmentioning

confidence: 98%

Optimal arrangements of classical and quantum states with limited purity

Bodmann

King

2019

J. London Math. Soc.

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“…Returning to the more general family of relaxations of the cut polytope described in Definition 1.2, the main question we propose for investigation is the following. Gershgorin NERF Our Bound Steiner: Affine [17] q 3 + 2q 2 q 2 + q q 2 + q q 2 + q − 1 q 2 + q Steiner: Projective [17] q 3 + 3q 2 + 3q + 2 q 2 + q + 1 q 2 + 2q + 2 q 2 + 3q + 1 q 2 + 3q + 2 Polyphase BIBD [13] q 3 + 1 q 2 − q + 1 q 2 + 1 q 2 + q − 1 q 2 + q Hyperovals [16] q 3 + q 2 − q q 2 + q − 1 q 2 q 2 − q + 3 q 2 − q + 4 Table 1: Spark Lower Bound Comparison. We tabulate three lower bounds on the spark of ETFs of N vectors in R r formed as the Naimark complements of ETFs belonging to infinite families with N ∼ r 3/2 .…”

Section: Open Problemsmentioning

confidence: 99%

Sum-of-Squares Optimization and the Sparsity Structure of Equiangular Tight Frames

Bandeira

Kunisky

2019

2019 13th International Conference on Sampling Theory and Applications (SampTA)

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Equiangular tight frames (ETFs) may be used to construct examples of feasible points for semidefinite programs arising in sum-of-squares (SOS) optimization. We show how generalizing the calculations in a recent work of the authors' that explored this connection also yields new bounds on the sparsity of (both real and complex) ETFs. One corollary shows that Steiner ETFs corresponding to finite projective planes are optimally sparse in the sense of achieving tightness in a matrix inequality controlling overlaps between sparsity patterns of distinct rows of the synthesis matrix. We also formulate several natural open problems concerning further generalizations of our technique.The v i form an equiangular tight frame (ETF) if moreover there exists α ∈ [0, 1] such thatWhen moreover v i ∈ R r , we have X ∈ E N , and such points form an interesting subset of the elliptope's boundary. In both the real and complex cases, UNTFs and ETFs have been studied in great detail previously (e.g. [3,4,27,5,15]).The set E N , however, is only the first of a sequence of tightening relaxations of C N , described by the sum-of-squares (SOS) hierarchy. This gives sets we call degree d generalized elliptopes E N d for positive even integers d.where, identifying indices of Y with elements of [N ] d/2 , the following conditions hold.

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“…Here the analysis operator Φ * : H → C N extends to an operator Φ * : K → C N and {ϕ n } n∈N is a tight frame for its span precisely when ΦΦ * x = Cx for all x ∈ H = span({ϕ n } n∈N ) = C(Φ). As shown in [24], this is equivalent to having either ΦΦ * Φ = CΦ, (ΦΦ * ) 2 = CΦΦ * or (Φ * Φ) 2 = CΦ * Φ. In particular, {ϕ n } n∈N is a C-tight frame for some D-dimensional space if and only if its Gram matrix Φ * Φ has eigenvalues C and 0 with multiplicity D and N − D, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…Meanwhile, Steiner ETFs arise from balanced incomplete block designs [26,25]. This construction has recently been generalized to yield new infinite families of ETFs arising from projective planes that contain hyperovals, Steiner triple systems, and group divisible designs [24,21,19].…”

mentioning

confidence: 99%

Harmonic equiangular tight frames comprised of regular simplices

Fickus

Schmitt

2020

Linear Algebra and its Applications

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An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs which equate to difference sets in finite abelian groups. Recently, it was shown that some harmonic ETFs are comprised of regular simplices. In this paper, we continue the investigation into these special harmonic ETFs. We begin by characterizing when the subspaces that are spanned by the ETF's regular simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of optimal packing in a Grassmannian space. We shall see that every difference set that produces an EITFF in this way also yields a complex circulant conference matrix. Next, we consider a subclass of these difference sets that can be factored in terms of a smaller difference set and a relative difference set. It turns out that these relative difference sets lend themselves to a second, related and yet distinct, construction of complex circulant conference matrices. Finally, we provide explicit infinite families of ETFs to which this theory applies.

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Equiangular Tight Frames From Hyperovals

Cited by 63 publications

References 37 publications

Optimal arrangements of classical and quantum states with limited purity

Optimal arrangements of classical and quantum states with limited purity

Sum-of-Squares Optimization and the Sparsity Structure of Equiangular Tight Frames

Harmonic equiangular tight frames comprised of regular simplices

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