When Complementary Measurements Count the Same Classical Bit Twice: Counterexamples to CQC, ECQC, and Complementarity-Based Certification

• URL: http://arxiv.org/abs/2608.03828v2
• Date: Wed, 05 Aug 2026 13:38:42 GMT
• Title: When Complementary Measurements Count the Same Classical Bit Twice: Counterexamples to CQC, ECQC, and Complementarity-Based Certification
• Abstract summary: We show that a correlation that is classical in one basis should disappear in a complementary basis in higher dimensions.<n>In experiments and applications, the same low-overhead architecture can serve as a calibration test before a multibasis score is assigned quantum meaning.
• Score: 3.3788644177253624
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $d\geq3$. 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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