Equiangular Subspaces in Euclidean Spaces

Balla, Igor; Sudakov, Benny

doi:10.1007/s00454-018-9972-5

Cited by 3 publications

(2 citation statements)

References 22 publications

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“…5. We expect that our approach should extend beyond lines to higher-order equiangular subspaces with respect to different notions of angle, which are described in [6]. In particular, we predict that our methods can be generalized to equiangular subspaces with respect to the chordal distance.…”

Section: Discussionmentioning

confidence: 94%

Equiangular lines via matrix projection

Balla¹

2021

Preprint

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In 1973, Lemmens and Seidel posed the problem of determining N R α (r), the maximum number of equiangular lines in R r with common angle arccos(α). Recently, this question has been almost completely settled in the case where r is large relative to 1/α, with the approach relying on Ramsey's theorem. In this paper, we use orthogonal projections of matrices with respect to the Frobenius inner product in order to overcome this limitation, thereby obtaining upper bounds on N R α (r) which significantly improve on the only previously known universal bound of Glazyrin and Yu, as well as taking an important step towards determining N R α (r) exactly for all r, α. In particular, our results imply that Nα(r) = Θ(r) for α ≥ Ω(1/r 1/5 ).Our arguments rely on a new geometric inequality for equiangular lines in R r which is tight when the number of lines meets the absolute bound r+1 2 . Moreover, using the connection to graphs, we obtain lower bounds on the second eigenvalue of the adjacency matrix of a regular graph which are tight for strongly regular graphs corresponding to r+1 2 equiangular lines in R r . Our results only require that the spectral gap is less than half the number of vertices and can therefore be seen as an extension of the Alon-Boppana theorem to dense graphs.Generalizing to C, we also obtain the first universal bound on N C α (r), the maximum number of complex equiangular lines in C r with common Hermitian angle arccos(α). In particular, we prove an inequality for complex equiangular lines in C r which is tight if the number of lines meets the absolute bound r 2 and may be of independent interest in quantum theory. Additionally, we use our projection method to obtain an improvement to Welch's bound.

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Section: Discussionmentioning

confidence: 94%

Equiangular lines via matrix projection

Balla¹

2021

Preprint

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“…The classical notions of canonical angles and isoclinic subspaces have played a role in Euclidean geometry, and in matrix and operator theory and beyond for over a century [10,1,3,9,23,24]. On the other hand, quantum information theory is relatively new, with roots going back several decades but only emerging as a formal field of study over the past quarter century or so [19].…”

Section: Introductionmentioning

confidence: 99%

Isoclinic Subspaces and Quantum Error Correction

Kribs¹,

Mammarella²,

Pereira³

2019

Preprint

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We exhibit equivalent conditions for subspaces of an inner product space to be isoclinic, including a characterization based on the classical notion of canonical angles. We identify a connection with quantum error correction, showing that every quantum error correcting code is associated with a family of isoclinic subspaces, and we prove a converse for pairs of such subspaces. We also show how the canonical angles for isoclinic subspaces arise in the structure of the higher rank numerical ranges of the corresponding orthogonal projections.

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An extension of Bravyi-Smolin’s construction for UMEBs

Levick

Rahaman

2021

Quantum Inf Process

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Equiangular Subspaces in Euclidean Spaces

Cited by 3 publications

References 22 publications

Equiangular lines via matrix projection

Equiangular lines via matrix projection

Isoclinic Subspaces and Quantum Error Correction

An extension of Bravyi-Smolin’s construction for UMEBs

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