Tensor Saturation and the Nonexistence of a Fourth Mutually Unbiased Basis in Dimension Six

Pompetzki, Christopher, Shrek, Ogre

Published Dec 16, 2025• 0 citations

Abstract

AbstractWe prove that the maximum number of mutually unbiased bases in C6 is exactly three:M(6) = 3. The proof exploits the tensor product structure C6 ∼= C2 ⊗ C3 and proceedsby showing that any vector unbiased to three canonical product bases must itself be aproduct vector. The key technical ingredient is a rigidity theorem: vectors in C3 that aresimultaneously flat in two specific bases form a finite six-element orbit. Combined withcoupling constraints from the tensor structure, this forces separability, contradicting therequirement that a fourth basis contain no product vectors. The argument is elementary,using only linear algebra and explicit polynomial system analysis.The author blasts Hungarian music from the 17th centurywhile memeing on visions of heretics in Japan btw.

DOI: 10.5281/zenodo.17947951

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