Equiangular spherical codes in quantum cryptography

Equiangular spherical codes in quantum cryptography

Equiangular Lines and Spherical Codes in Euclidean Space

A family of lines through the origin in Euclidean space is called equiangular if any pair of lines defines the same angle. The problem of estimating the maximum cardinality of such a family in R was extensively studied for the last 70 years. Motivated by a question of Lemmens and Seidel from 1973, in this paper we prove that for every fixed angle θ and sufficiently large n there are at most 2n−...

Bounds on Equiangular Lines and on Related Spherical Codes

An L-spherical code is a set of Euclidean unit vectors whose pairwise inner products belong to the set L. We show, for a fixed α, β > 0, that the size of any [−1,−β] ∪ {α}-spherical code is at most linear in the dimension. In particular, this bound applies to sets of lines such that every two are at a fixed angle to each another.

Codes for Key Generation in Quantum Cryptography

As an alternative to the usual key generation by two-way communication in schemes for quantum cryptography, we consider codes for key generation by one-way communication. We study codes that could be applied to the raw key sequences that are ideally obtained in recently proposed scenarios for quantum key distribution, which can be regarded as communication through symmetric four-letter channels.

New Bounds for Equiangular Lines and Spherical Two-Distance Sets

The set of points in a metric space is called an s-distance set if pairwise distances between these points admit only s distinct values. Two-distance spherical sets with the set of scalar products {α,−α}, α ∈ [0, 1), are called equiangular. The problem of determining the maximal size of s-distance sets in various spaces has a long history in mathematics. We determine a new method of bounding th...