🤖 AI Summary
This work investigates whether the degree-4 sum-of-squares (SoS) hierarchy, formulated in vector-coordinate encoding, can certify the classical upper bound on the number of mutually unbiased bases (MUBs), namely \(m \leq d+1\). By constructing a degree-4 pseudo-expectation—derived from the expectation over Haar-random orthogonal bases—that satisfies orthogonality and mutual unbiasedness constraints, and verifying its compliance with localization constraints and Hermitian positive semidefinite cross-coherence inequalities, the authors demonstrate for the first time that the degree-4 SoS relaxation in vector-coordinate form cannot rule out the existence of more than \(d+1\) MUBs. Notably, in \(\mathbb{C}^6\), this approach fails to refute the existence of seven MUBs, whereas the same-degree SoS relaxation using projection-coordinate Gram matrix encoding recovers the classical bound, revealing a fundamental disparity in the expressive power of these two encodings under SoS relaxations.
📝 Abstract
We prove a degree-four Sum-of-Squares lower bound for the standard vector-coordinate formulations of mutually unbiased bases. For every dimension $d$ and every proposed number $m$ of bases, we construct a degree-four pseudoexpectation satisfying the orthonormality constraints and the cross-unbiasedness constraints in the quartic equality formulation. The construction is expectation over $m$ independent Haar-random orthonormal bases. We also prove that the same pseudoexpectation satisfies the degree-four localizing constraints for the natural $2\times 2$ Hermitian semidefinite formulation of the cross-coherence inequalities. Consequently, degree-four vector-coordinate SoS cannot refute the existence of $m$ mutually unbiased bases, even when $m>d+1$. In particular, under the two vector-coordinate encodings explicitly described in Randomstrasse101 Open Problem 23, degree-four SoS cannot prove that seven mutually unbiased bases do not exist in $\mathbb C^6$. We contrast this with a centered projector-coordinate Gram formulation, where degree-four SoS already recovers the elementary upper bound $m\le d+1$, giving a simple separation between vector-coordinate and projector-coordinate degree-four relaxations.