Finite two-distance tight frames

Barg, Alexander; Glazyrin, Alexey; Okoudjou, Kasso A.; Yu, Wei

doi:10.1016/j.laa.2015.02.020

Cited by 42 publications

(54 citation statements)

References 24 publications

(44 reference statements)

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“…In the previous proof, our certificateĴ + P is the Gram matrix of a 2-distance tight frame, and such objects were recently studied in [10]. While the previous result indicates that ETFs can be used to build smaller packings, the following result uses ETFs to build larger packings: Corollary 16.…”

Section: Certifying Strongly Locally Optimal Packingsmentioning

confidence: 93%

Packings in Real Projective Spaces

Fickus¹,

Jasper²,

Mixon³

2018

SIAM J. Appl. Algebra Geometry

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This paper applies techniques from algebraic and differential geometry to determine how to best pack points in real projective spaces. We present a computer-assisted proof of the optimality of a particular 6-packing in RP 3 , we introduce a linear-time constant-factor approximation algorithm for packing in the so-called Gerzon range, and we provide local optimality certificates for two infinite families of packings. Finally, we present perfected versions of various putatively optimal packings from Sloane's online database, along with a handful of infinite families they suggest, and we prove that these packings enjoy a certain weak notion of optimality.the correlation | x, y |; thanks to this relationship, our representation of points in RP d−1 by unit vectors in R d is not problematic. The coherence of an n-packing Φ = {ϕ i } i∈[n] is given byAn n-packing Φ in RP d−1 is optimal if µ(Φ) ≤ µ(Ψ) for all n-packings Ψ in RP d−1 ; such packings necessarily exist by compactness. Optimal packings are called Grassmannian frames in the finite frames literature [71], but we avoid this terminology here for the sake of clarity. Optimal packings in the d = 2 case are trivial: RP 1 is isometric to the circle, and so optimal packings amount to regularly spaced points by the pigeonhole principle [11]. Packing is more difficult in higher dimensions. In [78], Welch introduced a useful lower bound:

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Section: Certifying Strongly Locally Optimal Packingsmentioning

confidence: 93%

Packings in Real Projective Spaces

Fickus¹,

Jasper²,

Mixon³

2018

SIAM J. Appl. Algebra Geometry

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“…One is naturally lead to ask if, given a FUNTF {f k } N k=1 ⊂ C d , can one find a unitary traceless d × d matrix U such that {f k } N k=1 and {U f k } N k=1 are companion ETFs. Before we answer this question in some special cases, we note that if {f j } N j=1 is an equiangular FUNTF for C d , the set of N 2 d × d matrices defined by {f j ⊗ f k = f j f * k } N j,k=1 forms a two distance tight frame for C d×d under the Hilbert Schmidt inner product, [15].…”

Section: Companion Etf In Prime Dimensionsmentioning

confidence: 99%

Equiangular quantum key distribution in more than two dimensions

Balu¹,

Koprowski²,

Okoudjou³

et al. 2019

J. Phys. A: Math. Theor.

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We extend the spherical code based key distribution protocols to qudits with dimensions 4 and 16 by constructing equiangular frames and their companions. We provide methods for equiangular frames in arbitrary dimensions for Alice to use and the companion frames, that has one antipode to eliminate one of the possibilities, made up of qudits with N = 4, 16 as part of Bob's code. Nonorthogonal bases that form positive operator valued measures can be constructed using the tools of frames (overcomplete bases of a Hilbert space) and here we apply them to key distribution that are robust due to large size of the bases making it hard for eavesdropping. We demonstrate a method to construct a companion frame for an equiangular tight frame for C p−1 generated from the discrete Fourier transform, where p is any odd prime. The security analysis is based on the assumption restricting possible attacks to intercept/resend scenario highlighting the advantages of a qudit over qubit-based protocols.

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“…Though these necessary conditions are not sufficient [27], they are asymptotically sufficient in two distinct ways: for any fixed K ≥ 2 and M ≥ 1, there exists U 0 = U 0 (K, M ) such that a K-GDD of type M U exists for all U ≥ U 0 such that (7) is satisfied [13,37]; for any fixed U ≥ K ≥ 2, there exists M 0 = M 0 (K, U ) such that a K-GDD of type M U exists for all M ≥ M 0 such that (7) is satisfied [43]. In the M = 1 and U = K cases, these facts reduce to more classical asymptotic existence results for BIBDs and MOLS, respectively.…”

Section: Group Divisible Designsmentioning

confidence: 99%

“…where S = 3, 5 (real maximal type [23]); (i) (K, S) = (4, 7), (6, 11), (7,13), (8,15), (10,5) (various other ETFs [23]).…”

Section: Negative Equiangular Tight Framesmentioning

confidence: 99%

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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Finite two-distance tight frames

Cited by 42 publications

References 24 publications

Packings in Real Projective Spaces

Packings in Real Projective Spaces

Equiangular quantum key distribution in more than two dimensions

Equiangular tight frames from group divisible designs

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