Polyphase equiangular tight frames and abelian generalized quadrangles

Fickus, Matthew; Jasper, John; Mixon, Dustin G.; Peterson, Janice; Watson, Cody E.

doi:10.1016/j.acha.2017.11.007

Cited by 20 publications

(59 citation statements)

References 43 publications

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“…Instead of thinking about signing the rows of

F_{3}

, as in the construction of Steiner equiangular tight frames, one may consider this construction to arise in part from signing the elements of the incidence matrix in order to induce linear dependencies. Following Example 3.2 in , we see that while

A

is full rank,

\tilde{A} = (\begin{matrix} 1 & 0 & - 1 \\ - 1 & 1 & 0 \\ 0 & - 1 & 1 \end{matrix})

is not. In , the authors phase incidence matrices (without additionally expanding them with rows of a unitary matrix) to construct equiangular tight frames; however, their approach is different since the constructed vectors are only frames for their spans.…”

Section: Gabor–steiner Equiangular Tight Framesmentioning

confidence: 75%

“…Instead of thinking about signing the rows of F 3 , as in the construction of Steiner equiangular tight frames, one may consider this construction to arise in part from signing the elements of the incidence matrix in order to induce linear dependencies. Following Example 3.2 in [34], we see that while A is full rank,…”

Section: Gabor-steiner Equiangular Tight Framesmentioning

confidence: 89%

“…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%

“…These are called phased BIBD ETFs . We analyze the Naimark complement of the specific construction in [, Theorem 5.1] (arising from a so‐called

(q^{2}, q, q)

‐polyphase BIBD ETF ), which has the same parameters as Gabor–Steiner ETFs for groups of size odd prime power

q = p^{s + 1}

. Let

χ

be a non‐trivial character over the additive group of

G F (q)

and let

false(k, κ false), false(\overset{k}{true}, \overset{κ}{true} false) \in G F {(q)}^{2}

.…”

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

confidence: 99%

“…is not. In [34], the authors phase incidence matrices (without additionally expanding them with rows of a unitary matrix) to construct equiangular tight frames; however, their approach is different since the constructed vectors are only frames for their spans.…”

Section: Gabor-steiner Equiangular Tight Framesmentioning

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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“…Instead of thinking about signing the rows of

F_{3}

A

is full rank,

\tilde{A} = (\begin{matrix} 1 & 0 & - 1 \\ - 1 & 1 & 0 \\ 0 & - 1 & 1 \end{matrix})

Section: Gabor–steiner Equiangular Tight Framesmentioning

confidence: 75%

Section: Gabor-steiner Equiangular Tight Framesmentioning

confidence: 89%

“…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%

“…These are called phased BIBD ETFs . We analyze the Naimark complement of the specific construction in [, Theorem 5.1] (arising from a so‐called

(q^{2}, q, q)

‐polyphase BIBD ETF ), which has the same parameters as Gabor–Steiner ETFs for groups of size odd prime power

q = p^{s + 1}

. Let

χ

be a non‐trivial character over the additive group of

G F (q)

and let

false(k, κ false), false(\overset{k}{true}, \overset{κ}{true} false) \in G F {(q)}^{2}

.…”

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

confidence: 99%

Section: Gabor-steiner Equiangular Tight Framesmentioning

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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Equiangular tight frames from group divisible designs

Fickus

Jasper

2018

Des. Codes Cryptogr.

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An equiangular tight frame (ETF) is a type of optimal packing of lines in a real or complex Hilbert space. In the complex case, the existence of an ETF of a given size remains an open problem in many cases. In this paper, we observe that many of the known constructions of ETFs are of one of two types. We further provide a new method for combining a given ETF of one of these two types with an appropriate group divisible design (GDD) in order to produce a larger ETF of the same type. By applying this method to known families of ETFs and GDDs, we obtain several new infinite families of ETFs. The real instances of these ETFs correspond to several new infinite families of strongly regular graphs. Our approach was inspired by a seminal paper of Davis and Jedwab which both unified and generalized McFarland and Spence difference sets. We provide combinatorial analogs of their algebraic results, unifying Steiner ETFs with hyperoval ETFs and Tremain ETFs.

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