Equiangular tight frames that contain regular simplices

Fickus, Matthew; Jasper, John; King, Emily J.; Mixon, Dustin G.

doi:10.1016/j.laa.2018.06.004

Cited by 26 publications

(58 citation statements)

References 63 publications

(283 reference statements)

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“…There seems to be evidence that a stronger result than Theorem holds, namely, that for

p

an odd prime,

\tilde{A G} false(2, p^{2} false)

contains no copies of

A G (2, p^{2})

. We note that by applying [, Theorem 5.2] to [, Theorem 5.1] over a finite field of order

p^{2}

, we may obtain an equiangular tight frame of

p^{4}

vectors spanning a space of dimension

p^{2} false(p^{2} - 1 false) / 2

which has a binder which contains

A G (2, p^{2})

. However, one may verify via triple products that these equiangular tight frames are not switching equivalent.…”

Section: A Family Of Gabor Equiangular Tight Frames Associated With Bmentioning

confidence: 99%

“…The following theorem is a specific application of [, Theorem 6.2], which generalizes [, Section 2.4], but we will prove it here directly with Lemma and state it within the context of spectrahedron arrangements. Theorem Let

m = (m_{0}, \dots, m_{s})

be a vector of odd integers

⩾ 3

with

false| m false| = \prod_{ℓ = 0}^{s} m_{ℓ}

and construct

G (m)

with binder

scriptB

.…”

Section: Spectrahedral Arrangements From Binders Of Equiangular Tightmentioning

confidence: 99%

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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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“…There seems to be evidence that a stronger result than Theorem holds, namely, that for

p

an odd prime,

\tilde{A G} false(2, p^{2} false)

contains no copies of

A G (2, p^{2})

. We note that by applying [, Theorem 5.2] to [, Theorem 5.1] over a finite field of order

p^{2}

, we may obtain an equiangular tight frame of

p^{4}

vectors spanning a space of dimension

p^{2} false(p^{2} - 1 false) / 2

which has a binder which contains

A G (2, p^{2})

. However, one may verify via triple products that these equiangular tight frames are not switching equivalent.…”

Section: A Family Of Gabor Equiangular Tight Frames Associated With Bmentioning

confidence: 99%

m = (m_{0}, \dots, m_{s})

be a vector of odd integers

⩾ 3

with

false| m false| = \prod_{ℓ = 0}^{s} m_{ℓ}

and construct

G (m)

with binder

scriptB

.…”

Section: Spectrahedral Arrangements From Binders Of Equiangular Tightmentioning

confidence: 99%

Optimal arrangements of classical and quantum states with limited purity

Bodmann

King

2019

J. London Math. Soc.

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“…In Theorem 3.5, we then characterize when the subspaces spanned by these simplices form a special type of ECTFF known as an equi-isoclinic tight fusion frame (EITFF). This occurs for some, but not all, of the ETFs considered in [20]. We further show that every difference set that produces an EITFF in this way also yields a complex circulant conference matrix C, namely an (S + 1) × (S + 1) circulant matrix whose diagonal entries are zero, whose off-diagonal entries are unimodular and for which C * C = SI.…”

mentioning

confidence: 76%

“…It was recently shown that certain harmonic ETFs are comprised of regular simplices [20]. To see this from basic principles, let D be a D-element difference set in an abelian group G of order G,…”

Section: Fine Difference Setsmentioning

confidence: 99%

“…Every harmonic ETF arising from a McFarland difference set is known to be unitarily equivalent to a Steiner ETF, and so also has this structure [31]. In a recent paper [20], it was shown that other harmonic ETFs, including those arising from the complements of certain Singer and twin prime power difference sets, are comprised of regular simplices despite not being unitarily equivalent to any Steiner ETF. There, it was further shown that when an ETF is comprised of regular simplices, the subspaces spanned by these simplices form a particular type of optimal packing in Grassmannian space known as an equi-chordal tight fusion frame (ECTFF), achieving the simplex bound of [12].…”

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confidence: 99%

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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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