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We consider the random matrix obtained by picking vectors randomly from alarge collection of mutually unbiased bases of $\mathbb{C}^n$, and prove thatthe spectral distribution converges to the Marchenko-Pastur law. This showsthat vectors in mutually unbiased bases behave like random vectors. Thisphenomenon is similar to that of binary linear codes of dual distance at least5, which was studied in previous work.

Spectral distribution of random matrices from Mutually Unbiased Bases