Optimal Line Packings From Finite Group actions

Iverson, Joseph W.; Jasper, John; Mixon, Dustin G.

doi:10.1017/fms.2019.48

Cited by 21 publications

(28 citation statements)

References 45 publications

(72 reference statements)

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“…Thus, the triple products arising from the construction in (or, more precisely, the Naimark complement of the equiangular tight frames constructed in those papers) with

ζ_{p}

are the same as the triple products for the Gabor–Steiner equiangular tight frame

G (p)

constructed in Definition over

Z_{p}

with primitive root

ζ_{p}^{- 2}

; hence, by Theorem , they are switching equivalent. Another equivalent construction appears in [, Theorem 6.4]. Namely, let

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

be a vector of odd integers and

G = \oplus_{ℓ = 0}^{s} {double-struckZ}_{m_{ℓ}}

.…”

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

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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“…Thus, the triple products arising from the construction in (or, more precisely, the Naimark complement of the equiangular tight frames constructed in those papers) with

ζ_{p}

are the same as the triple products for the Gabor–Steiner equiangular tight frame

G (p)

constructed in Definition over

Z_{p}

with primitive root

ζ_{p}^{- 2}

; hence, by Theorem , they are switching equivalent. Another equivalent construction appears in [, Theorem 6.4]. Namely, let

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

be a vector of odd integers and

G = \oplus_{ℓ = 0}^{s} {double-struckZ}_{m_{ℓ}}

.…”

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

confidence: 99%

“…These frames are interesting in their own right. In certain cases, they are equivalent to known indirect constructions which leverage combinatorial designs or group actions but not both . We also provide the first complete characterization of the binders of an infinite class of equiangular tight frames.…”

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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“…In the case where G is abelian, the resulting ETFs correspond to difference sets in the combinatorial design literature [71,79,30]. The Heisenberg-Weyl group can be used to construct all known SICs [43], as well as an infinite family of non-SIC ETFs [53].…”

Section: The State Of Playmentioning

confidence: 99%

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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“…Recently, Iverson et al 15 studied supersaturated designs (frames) with small s max that can be obtained from permutation group actions, where the Gram matrices are G‐circulant for some permutation group G. On the other hand, Morales and Vega 16 classified k ‐circulant SSDs with s max ∈{2,4} up to isomorphism. In this case, the Gram matrix D T D is block

ℤ_{n}

‐circulant with k 2 blocks of ( n × n ) matrices, where n is the number of rows in D .…”

Section: Conclusion and A Future Research Directionmentioning

confidence: 99%

On the maximum number of columns for supersaturated designs with s _max ∈{1,3,5}

Morales

2020

Comp and Math Methods

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Cheng and Tang [Upper bounds on the number of columns in supersaturated designs. Biometrika, 88 (2001), 1169‐1174] provided upper bounds on the maximum number of columns, denoted by B(n,t), that can be accommodated in two‐symbol supersaturated designs for a given number, say n, of rows and a maximum correlation in absolute value, say t/n, between any two columns. Recently, Morales et al [On the maximum number of columns in supersaturated designs with smax=2, J. Combin. Des., 27 (2019), 448‐472] proved that B(n,2)=n+1 for n=14,18,22,30. However, from the 35 lower bounds for B(n,t) provided by Cheng and Tang, only 11 supersaturated designs are known to satisfy these bounds. In this article, by performing a computer search, we show that B(9,1)=7, B(13,1)=12, B(17,1)=15, B(21,1)=19, B(7,3)=15, B(9,3)=12, B(11,3)=B(13,3)=15, B(11,5)=66, and B(15,3)=17. Likewise, our search produces supersaturated designs that achieve these maximums. Each of these exact values was previously unknown.

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Optimal Line Packings From Finite Group actions

Cited by 21 publications

References 45 publications

Optimal arrangements of classical and quantum states with limited purity

Optimal arrangements of classical and quantum states with limited purity

Packings in Real Projective Spaces

On the maximum number of columns for supersaturated designs with s _max ∈{1,3,5}

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Optimal Line Packings From Finite Group actions

Cited by 21 publications

References 45 publications

Optimal arrangements of classical and quantum states with limited purity

Optimal arrangements of classical and quantum states with limited purity

Packings in Real Projective Spaces

On the maximum number of columns for supersaturated designs with s max ∈{1,3,5}

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Product

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On the maximum number of columns for supersaturated designs with s _max ∈{1,3,5}