Achieving the orthoplex bound and constructing weighted complex projective 2-designs with Singer sets

Bodmann, Bernhard G.; Haas, John I.

doi:10.1016/j.laa.2016.09.005

Cited by 33 publications

(44 citation statements)

References 36 publications

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“…Among the results that we discuss in the classical setting are BIBDs as the support set of distance maximizers. This was realized in a prior paper in a special case in which probability vectors had a support that was half the number of outcomes/vertices [14]. Here, in Corollary 2.24, we deduce that a family of designs on q vertices, where q is a prime power, n = q(q m − 1)/(q − 1) blocks of size k = q m−1 , intersecting in at most q m−1 vertices gives rise to n probability vectors that saturate a bound by Rankin for packings in a simplex while purity is limited by 1/k.…”

Section: Introductionmentioning

confidence: 99%

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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Section: Introductionmentioning

confidence: 99%

Optimal arrangements of classical and quantum states with limited purity

Bodmann

King

2019

J. London Math. Soc.

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“…Using Theorem 2.2, one can determine a set that is optimal with respect to {f i } d+1 i=1 in the sense of Definition 2.1. The bound in (2.1) is similar to the orthoplex bound [31].…”

Section: Constructing Low Coherence Untfs With Block Designsmentioning

confidence: 54%

Low coherence unit norm tight frames

Datta

Oldroyd

2018

Linear and Multilinear Algebra

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Equiangular tight frames (ETFs) have found significant applications in signal processing and coding theory due to their robustness to noise and transmission losses. ETFs are characterized by the fact that the coherence between any two distinct vectors is equal to the Welch bound. This guarantees that the maximum coherence between pairs of vectors is minimized. Despite their usefulness and widespread applications, ETFs of a given size N are only guaranteed to exist in R d or C d if N = d + 1. This leads to the problem of finding approximations of ETFs of N vectors in R d or C d where N > d + 1. To be more precise, one wishes to construct a unit norm tight frame (UNTF) such that the maximum coherence between distinct vectors of this frame is as close to the Welch bound as possible. In this paper low coherence UNTFs in R d are constructed by adding a strategically chosen set of vectors called an optimal set to an existing ETF of d + 1 vectors. In order to do so, combinatorial objects called block designs are used. Estimates are provided on the maximum coherence between distinct vectors of this low coherence UNTF. It is shown that for certain block designs, the constructed UNTF attains the smallest possible maximum coherence between pairs of vectors among all UNTFs containing the starting ETF of d + 1 vectors. This is particularly desirable if there does not exist a set of the same size for which the Welch bound is attained.

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“…Again assuming {ϕ j } j∈K is a regular simplex, all that remains to be shown is that any sequence of scalars {c j } j∈K that satisfies (12) is a Naimark complement of it. From above, we know the aforementioned sequence {ĉ j } j∈K is a Naimark complement for {ϕ j } j∈K and so satisfies (12). If {c j } j∈K also satisfies (12), then both sequences lie in the null space of the synthesis operator Φ K of the regular simplex.…”

Section: Finding the Regular Simplices Contained In An Equiangular Timentioning

confidence: 98%

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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Achieving the orthoplex bound and constructing weighted complex projective 2-designs with Singer sets

Cited by 33 publications

References 36 publications

Optimal arrangements of classical and quantum states with limited purity

Optimal arrangements of classical and quantum states with limited purity

Low coherence unit norm tight frames

Equiangular tight frames that contain regular simplices

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