Spectral distribution of random matrices from mutually unbiased bases

Abstract

<p style='text-indent:20px;'>We consider the random matrix obtained by picking vectors randomly from a large collection of mutually unbiased bases of <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{C}^n $\end{document}</tex-math></inline-formula>, 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 ([<xref ref-type="bibr" rid="b33">33</xref>]), which was studied in previous work.</p>