Qubit Incompatibility Probability
Abstract
The qubit incompatibility probability is strictly less than one quarter.
Theorem 1.1 (One-quarter conjecture refuted).
Proof. Machine-checked in Lean as D5/S3/Quantum/Measurement/QubitIncompatibilityProbability.result (✓ std3). ∎
Resolves. Problems/zhang-et-al-2019-qubit-incompatibility-probability (refuted) by D5/S3/Quantum/Measurement/QubitIncompatibilityProbability.result.
Source. Repository-derived.
Acknowledgement. Lin Zhang; Hua Xiang; Xiao Li-Jost; Shao-Ming Fei (2019). Incompatibility probability of random quantum measurements. DOI: 10.1103/PhysRevE.100.062139. URL: https://arxiv.org/abs/1912.12321.
Commentary.
The exact probability is nine times a four-dimensional cube integral. The inequality sqrt(1-z) ≤ 1-z/2-z²/8 on [0,1], together with three explicit moments, gives the upper bound 5910272/23648625 < 1/4. Thus the probability in Section III.C of Zhang et al. differs from one quarter.