Nearly orthogonal vectors and small antipodal spherical codes
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NEARLY ORTHOGONAL VECTORS AND SMALL ANTIPODAL SPHERICAL CODES BY
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Boris Bukh and Christopher Cox∗∗ Department of Mathematical Sciences, Carnegie Mellon University Pittsburgh, PA 15213, USA e-mail: [email protected], [email protected]
ABSTRACT
How can d+k vectors in Rd be arranged so that they are as close to orthogonal as possible? In particular, define θ(d, k) := minX maxx=y∈X |x, y| where the minimum is taken over all collections of d + k unit vectors X ⊆ Rd . In this paper, we focus on the case where k is fixed and d → ∞. In establishing bounds on θ(d, k), we find an intimate connection to the existence of systems of k+1 equiangular lines in Rk . Using this con2 nection, we are able to pin down θ(d, k) whenever k ∈ {1, 2, 3, 7, 23} and establish asymptotics for general k. The main tool is an upper bound on Ex,y∼µ |x, y| whenever μ is an isotropic probability mass on Rk , which may be of independent interest. Our results translate naturally to the analogous question in Cd . In this case, the question relates to the existence of systems of k 2 equiangular lines in Ck , also known as SIC-POVM in physics literature.
1. Introduction How can a given number of points be arranged on a sphere in Rd so that they are as far from each other as possible? This is a basic problem in coding theory; for example, the book [13] is devoted to this problem exclusively. Such point arrangements are called spherical codes. Most constructions of spherical codes ∗ Supported in part by Sloan Research Fellowship and by U.S. taxpayers through
NSF CAREER grant DMS-1555149. ∗∗ Supported in part by U.S. taxpayers through NSF CAREER grant DMS-1555149.
Received April 20, 2018 and in revised form July 16, 2019
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B. BUKH AND C. COX
Isr. J. Math.
are symmetric. Here we consider the antipodal codes, in which the points come in pairs x, −x. In other words, we seek arrangements of d + k unit vectors in Rd so that they are as close to orthogonal as possible. An alternative point of view is that these are codes in the projective space RPd−1 ; for example, see [9]. We focus on the case when k is small. As we will see, this question relates to the problem of the existence of large families of equiangular lines in Rk . Similarly, the analogous question for unit vectors in Cd relates to equiangular lines in Ck , which are the mathematical underpinning of symmetric informationally complete measurements in quantum theory [25]. Because of this, we elect to treat the real and complex cases in parallel. Henceforth, we denote by H the underlying field, which can be either R or C. For H ∈ {R, C}, define the parameter θH (d, k) := min max |x, y|, X
x=y∈X
where the minimum is taken over all collections of d + k unit vectors X ⊆ Hd . In this paper, we prove bounds on θH (d, k) when k is fixed and d → ∞. For a collection of vectors X = {x1 , . . . , xn } ⊆ Hd , the Gram matrix is the matrix A ∈ Hn×n where Aij = xi , xj . It will be easier to work with the Gram matrices than with the vectors themselves. For a matrix A ∈ Hn×n , define off(A) :=
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