Spectral distribution of random matrices from Mutually Unbiased Bases

TL;DR

This paper proves that the spectral distribution of random matrices formed from vectors in mutually unbiased bases converges to the Marchenko-Pastur law, indicating these vectors behave like random vectors.

Contribution

It demonstrates that vectors from mutually unbiased bases exhibit spectral properties similar to random vectors, extending understanding of their statistical behavior.

Findings

Spectral distribution converges to Marchenko-Pastur law

Vectors in mutually unbiased bases behave like random vectors

Similar phenomena observed in binary linear codes

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, which was studied in previous work.