Spectral distribution of random matrices from mutually unbiased bases

Chin Hei Chan; Maosheng Xiong

doi:10.3934/amc.2022072

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2022, Volume 16, Issue 4: 721-732. Doi: 10.3934/amc.2022072

This issue Previous Article Möbius transformations and characterizations of hyper-bent functions from Dillon-like exponents with coefficients in extension fields Next Article Estimate of 4-adic complexity of unified quaternary sequences of length $ 2p $

Spectral distribution of random matrices from mutually unbiased bases

Chin Hei Chan and
Maosheng Xiong^,

Dept. of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

^*Corresponding author: Maosheng Xiong
^*Corresponding author: Maosheng Xiong

Received: March 2022

Revised: June 2022

Early access: September 2022

Published: November 2022

The second author is supported RGC grant number 16306520 from Hong Kong

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Abstract

We consider the random matrix obtained by picking vectors randomly from a large collection of mutually unbiased bases of $ \mathbb{C}^n $, and prove that the spectral distribution converges to the Marchenko-Pastur law. This shows that vectors in mutually unbiased bases behave like random vectors. This phenomenon is similar to that of binary linear codes of dual distance at least 5 ([33]), which was studied in previous work.

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Mathematics Subject Classification: Primary: 15B52, 94D99; Secondary: 60B20.

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