The cost of direction in quantum Bell certification

Abstract

Certifying a quantum bound can require more information than realizing the correlations that attain it. We show that this distinction becomes unbounded when a proof must condition on one party's measurement question. A fixed Bell scenario attained by one entangled qubit pair has a standard degree-two sum-of-squares certificate, yet the minimum degree in either conditioning direction grows as the inverse square root of the distance to a local endpoint. Choosing the better party does not remove this cost. The obstruction also survives approximate certification: bounding the quantum maximum to any fixed fractional accuracy relative to its Bell advantage requires the same diverging level. Thus a short standard proof cannot be transferred to a uniformly bounded conditioned level, even after relaxing exactness and optimising the direction. Mixing directions within a proof removes this family's obstruction, while a separate finite separation establishes its limits. Contact geometry explains the cost and connects it to a sharp precision law for randomness certification.