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.

翻译：我们证明了对于标准向量坐标表述的互不偏基，一个四阶和-平方下界。针对每个维度$d$和每个提议的基的数量$m$，我们构造了一个满足四次等式表述中的正交归一约束和互不偏约束的四阶伪期望。该构造是对$m$个独立Haar随机正交基的期望。我们还证明了相同的伪期望满足交叉相干不等式自然$2\times 2$厄米半正定表述的四阶局域化约束。因此，四阶向量坐标SOS无法反驳存在$m$个互不偏基，即使当$m>d+1$时。特别地，在Randomstrasse101开放问题23中明确描述的两种向量坐标编码下，四阶SOS无法证明在$\mathbb C^6$中不存在七个互不偏基。我们将此与一个中心投影坐标Gram表述进行对比，其中四阶SOS已经恢复了基本上界$m\le d+1$，从而在向量坐标和投影坐标四阶松弛之间给出了一个简单的区分。