Gabor fusion frames generated by difference sets

Bojarovska, Irena; Paternostro, Victoria

doi:10.1117/12.2186394

Cited by 6 publications

(12 citation statements)

References 26 publications

(49 reference statements)

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“…If

D

is a difference set

G

, then the submatrix

normalΦ

F_{m}

consisting of the rows corresponding to

D

and the columns corresponding to all of

\overset{G}{true}

generates an equiangular tight frame . Inspired by , one may write this as a group action by letting

ψ \in {double-struckC}^{| m |}

be the vector that is 1 on

D

and 0 on

G ∖ D

and letting

\{M_{m}^{false(κ false)} : κ = (κ_{0}, \dots, κ_{s}) \in ⨁_{ℓ = 0}^{s} Z_{m_{ℓ}}\} ≅ G

act on

ψ

. 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.…”

Section: Gabor–steiner Equiangular Tight Framesmentioning

confidence: 99%

“…We note that in , Gabor frames are formed from the orbit of

1_{D}

under

σ ({double-struckZ}_{m} \times {double-struckZ}_{m})

, for some

m \in N

, where

D

is a so‐called difference set in

Z_{m}

and

1_{D}

is the vector that is 1 on

D

and 0 on

{double-struckZ}_{m} ∖ D

. The resulting frames are not equiangular but rather biangular ; however,

{β_{k} = false{σ (k, κ) 1_{D} : κ \in Z_{m} false} : k \in {double-struckZ}_{m}}

forms an optimal (equichordal)

(m, m)

‐spectrahedron arrangement of purity

1 / | D |

, where in general

| D | < m - 1

.…”

Section: Spectrahedral Arrangements From Binders Of Equiangular Tightmentioning

confidence: 99%

“…Then the Fourier matrix F m = F m0,...,ms is the character table of G. If D is a difference set [22] G, then the submatrix Φ of F m consisting of the rows corresponding to D and the columns corresponding to all of G generates an equiangular tight frame [27,28,41,71,79]. Inspired by [16], one may write this as a group action by letting ψ ∈ C |m| be the vector that is 1 on D and 0 on G\D and letting…”

Section: )mentioning

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.

show abstract

“…If

D

is a difference set

G

, then the submatrix

normalΦ

F_{m}

consisting of the rows corresponding to

D

and the columns corresponding to all of

\overset{G}{true}

generates an equiangular tight frame . Inspired by , one may write this as a group action by letting

ψ \in {double-struckC}^{| m |}

be the vector that is 1 on

D

and 0 on

G ∖ D

and letting

\{M_{m}^{false(κ false)} : κ = (κ_{0}, \dots, κ_{s}) \in ⨁_{ℓ = 0}^{s} Z_{m_{ℓ}}\} ≅ G

act on

ψ

. 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.…”

Section: Gabor–steiner Equiangular Tight Framesmentioning

confidence: 99%

“…We note that in , Gabor frames are formed from the orbit of

1_{D}

under

σ ({double-struckZ}_{m} \times {double-struckZ}_{m})

, for some

m \in N

, where

D

is a so‐called difference set in

Z_{m}

and

1_{D}

is the vector that is 1 on

D

and 0 on

{double-struckZ}_{m} ∖ D

. The resulting frames are not equiangular but rather biangular ; however,

{β_{k} = false{σ (k, κ) 1_{D} : κ \in Z_{m} false} : k \in {double-struckZ}_{m}}

forms an optimal (equichordal)

(m, m)

‐spectrahedron arrangement of purity

1 / | D |

, where in general

| D | < m - 1

.…”

Section: Spectrahedral Arrangements From Binders Of Equiangular Tightmentioning

confidence: 99%

Section: )mentioning

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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show abstract

“…Every composite difference set is an amalgam, and every amalgam is fine. In terms of the sets D g defined in (9), being fine equates to D g being empty when g ∈ H, and having equal cardinality otherwise. When D is an amalgam, we further have that each D g with g / ∈ H is a difference set for H. When D is a composite, we even further have that any two such D g are translates.…”

Section: Discussionmentioning

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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“…When n = 2d, this means that exactly half of all possible 3-element subsets of a real ETF {ϕ j } n j=1 satisfy (11) as evidenced, for example, by a list (14) of such subsets that arises in the d = 5 case.…”

Section: Constructing Naimark Complements From Binders With Combinatomentioning

confidence: 99%

Equiangular tight frames that contain regular simplices

Fickus

Jasper

King

et al. 2018

Linear Algebra and its Applications

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An equiangular tight frame (ETF) is a type of optimal packing of lines in Euclidean space. A regular simplex is a special type of ETF in which the number of vectors is one more than the dimension of the space they span. In this paper, we consider ETFs that contain a regular simplex, that is, have the property that a subset of its vectors forms a regular simplex. As we explain, such ETFs are characterized as those that achieve equality in a certain well-known bound from the theory of compressed sensing. We then consider the so-called binder of such an ETF, namely the set of all regular simplices that it contains. We provide a new algorithm for computing this binder in terms of products of entries of the ETF's Gram matrix. In certain circumstances, we show this binder can be used to produce a particularly elegant Naimark complement of the corresponding ETF. Other times, an ETF is a disjoint union of regular simplices, and we show this leads to a certain type of optimal packing of subspaces known as an equichordal tight fusion frame. We conclude by considering the extent to which these ideas can be applied to numerous known constructions of ETFs, including harmonic ETFs. Necessary integrality conditions on the existence of various types of ETFs are given in [58].Conference matrices, Hadamard matrices, Paley tournaments and quadratic residues are related, and lead to infinite families of ETFs whose redundancy n d is either nearly or exactly two [57,43,53,56]. Harmonic ETFs and Steiner ETFs offer much more freedom in choosing d and n. Harmonic ETFs are equivalent to difference sets in finite abelian groups [60,57,63,27]. Steiner ETFs arise from particular types of balanced incomplete block designs (BIBDs) [39,36]. Recent generalizations of Steiner ETFs have led to new infinite families of ETFs arising from projective planes that contain hyperovals [35] as well as from Steiner triple systems [31]. Another new family arises by generalizing the SRG construction of [38] to the complex setting [32].Many of these known constructions give ETFs that contain a regular simplex, and thus achieve equality in both (2) and (3). For example, every Steiner ETF is a disjoint union of regular simplices by construction. This property is also enjoyed by harmonic ETFs arising from McFarland difference sets, since it is known that they can be obtained by applying unitary operators to certain Steiner ETFs [45]. We study ETFs that contain regular simplices in general, and then explore the degree to which the ETFs constructed in [63,27,35,31,32] have this property.In particular, in the next section, we establish notation, discuss some known results that we will need later on, and elaborate on the connections between these ideas and compressed sensing. In Section 3, we show that an ETF achieves equality in (3) if and only if it contains a regular simplex (Theorem 3.1), and give a strong necessary condition on the existence of real ETFs that are full spark (Theorem 3.2). In the fourth section, we characterize regular simplices that are containe...

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Gabor fusion frames generated by difference sets

Cited by 6 publications

References 26 publications

Optimal arrangements of classical and quantum states with limited purity

Optimal arrangements of classical and quantum states with limited purity

Harmonic equiangular tight frames comprised of regular simplices

Equiangular tight frames that contain regular simplices

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