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.
Related
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BibTeX
@misc{wang2026cost,
title = {{The cost of direction in quantum Bell certification}},
author = {Fumin Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.10162},
eprint = {2609.10162},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10162},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The cost of direction in quantum Bell certification
AU - Fumin Wang
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2609.10162
UR - https://arxiv.org/abs/2609.10162
AB - 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.
ER -