Nearly orthogonal vectors and small antipodal spherical codes

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 maxxyX |〈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 here k is fixed and d → ∞. In establishing bounds on θ(d, k), we find an intimate connection to the existence of systems of \(\left(\begin{array}{c}k+1\\ 2\end{array}\right)\) equiangular lines in ℝk. Using this connection, 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 \(\mathbb{E}_{x,y\sim\mu}|\langle{x,y}\rangle|\) 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 k2 equiangular lines in ℂk, also known as SIC-POVM in physics literature.

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Correspondence to Boris Bukh.

Additional information

Added in a revision. The conjecture has since been resolved by Glazyrin [16, Theorem 4].

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.

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Bukh, B., Cox, C. Nearly orthogonal vectors and small antipodal spherical codes. Isr. J. Math. 238, 359–388 (2020). https://doi.org/10.1007/s11856-020-2027-7

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