When Complementary Measurements Count the Same Classical Bit Twice: Counterexamples to CQC, ECQC, and Complementarity-Based Certification
Abstract
Mutually unbiased measurements are commonly expected to expose independent facets of a quantum state: a correlation that is classical in one basis should disappear in a complementary basis. In higher dimensions, however, this intuition becomes particularly subtle because correlations recovered in different settings need not represent different information. To expose this loophole, we propose a two-branch classical null test: before the setting is chosen, a shared bit selects one of two orthogonal product preparations, producing a rank-two classical--classical state, and the candidate protocol then runs unchanged. Different settings can read the same bit through different outcome patterns. This two-branch classical architecture disproves the complementary-quantum correlation (CQC) conjecture in every dimension d≥3. A distinct rank-two classical--classical state disproves its complete-basis extension (ECQC) at d=7, with an overrun that grows without bound along prime dimensions. Its qutrit CQC instance also gives classical false positives for a proposed quantum-correlation measure and a proposed one-sided semi-device-independent steering criterion, and refutes a conditional-probability conjecture. The failures identify the missing requirement: information read in different settings must be nonredundant. In experiments and applications, the same low-overhead architecture can serve as a calibration test before a multibasis score is assigned quantum meaning.
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BibTeX
@misc{wang2026when,
title = {{When Complementary Measurements Count the Same Classical Bit Twice: Counterexamples to CQC, ECQC, and Complementarity-Based Certification}},
author = {Jinbo Wang and Qihang Wang and Kun Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.03828},
eprint = {2608.03828},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.03828},
}
RIS
TY - UNPB
TI - When Complementary Measurements Count the Same Classical Bit Twice: Counterexamples to CQC, ECQC, and Complementarity-Based Certification
AU - Jinbo Wang
AU - Qihang Wang
AU - Kun Chen
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2608.03828
UR - https://arxiv.org/abs/2608.03828
AB - Mutually unbiased measurements are commonly expected to expose independent facets of a quantum state: a correlation that is classical in one basis should disappear in a complementary basis. In higher dimensions, however, this intuition becomes particularly subtle because correlations recovered in different settings need not represent different information. To expose this loophole, we propose a two-branch classical null test: before the setting is chosen, a shared bit selects one of two orthogonal product preparations, producing a rank-two classical--classical state, and the candidate protocol then runs unchanged. Different settings can read the same bit through different outcome patterns. This two-branch classical architecture disproves the complementary-quantum correlation (CQC) conjecture in every dimension d≥3. A distinct rank-two classical--classical state disproves its complete-basis extension (ECQC) at d=7, with an overrun that grows without bound along prime dimensions. Its qutrit CQC instance also gives classical false positives for a proposed quantum-correlation measure and a proposed one-sided semi-device-independent steering criterion, and refutes a conditional-probability conjecture. The failures identify the missing requirement: information read in different settings must be nonredundant. In experiments and applications, the same low-overhead architecture can serve as a calibration test before a multibasis score is assigned quantum meaning.
ER -