One Relative Phase Orders Operational Thresholds of Noisy Bell Pairs
Abstract
Unlike the two-qubit entangled and CHSH-nonlocal sets, the steerable and Bell-nonlocal ones have no known general closed-form criterion. For any such operational ability whose failing states form a convex set closed under local unitaries, and for a Bell mixture with arbitrary noise, the X part of the noise gives an upper bound on the definitive threshold, the Bell weight above which the mixture attains the ability. That bound is the tightest that the seven of the fifteen Pauli expectation values fixing the X part can support. The same theorem shows that a state has the ability whenever its X part does, so partial tomography of a state can certify its ability. When noise is of X form, the mixture carries a relative phase between the noise and Bell coherences that cannot be gauged away by local unitaries, and it controls the threshold through interference. We prove a monotonicity law: the threshold is nondecreasing in this phase, whether or not a closed-form criterion for the ability exists.
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BibTeX
@misc{trinh2026relative,
title = {{One Relative Phase Orders Operational Thresholds of Noisy Bell Pairs}},
author = {Xuan Du Trinh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10034},
eprint = {2609.10034},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10034},
}
RIS
TY - UNPB
TI - One Relative Phase Orders Operational Thresholds of Noisy Bell Pairs
AU - Xuan Du Trinh
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10034
UR - https://arxiv.org/abs/2609.10034
AB - Unlike the two-qubit entangled and CHSH-nonlocal sets, the steerable and Bell-nonlocal ones have no known general closed-form criterion. For any such operational ability whose failing states form a convex set closed under local unitaries, and for a Bell mixture with arbitrary noise, the X part of the noise gives an upper bound on the definitive threshold, the Bell weight above which the mixture attains the ability. That bound is the tightest that the seven of the fifteen Pauli expectation values fixing the X part can support. The same theorem shows that a state has the ability whenever its X part does, so partial tomography of a state can certify its ability. When noise is of X form, the mixture carries a relative phase between the noise and Bell coherences that cannot be gauged away by local unitaries, and it controls the threshold through interference. We prove a monotonicity law: the threshold is nondecreasing in this phase, whether or not a closed-form criterion for the ability exists.
ER -