Operational thresholds of Bell mixtures with complex X noise

Abstract: Noisy Bell mixtures may carry entanglement, standard teleportation usefulness, projective-measurement steerability, and optimized Cavalcanti-Jones-Wiseman-Reid (CJWR) and Clauser-Horne-Shimony-Holt (CHSH) violations. If the noise already has an ability, the mixture may lose and later recover it at distinct boundary crossings as the Bell weight decreases. We characterize ability-absence intervals and definitive thresholds for mixtures of $|Φ<sup>+\rangle$ with arbitrary complex two-qubit $X$ noise. A closed-form singular-value flow of the Pauli correlation tensor gives the exact teleportation-useless, CJWR-satisfying, and CHSH-local intervals, while the positive partial transpose criterion gives the exact separable interval. At fixed populations and coherence magnitudes, these intervals widen as the relative phase between the $Φ$-block coherence of the noise and the Bell coherence increases from $0$ to $π$. The definitive thresholds form a universal chain from entanglement through teleportation usefulness and the three-setting CJWR witness to the common two-setting CJWR witness and CHSH-nonlocal threshold. Replacing teleportation usefulness by steerability gives a second chain, although their thresholds are not mutually ordered. For arbitrary pure two-qubit noise, the steerability thresholds in both directions for projective measurements and arbitrary positive operator-valued measures equal the entanglement threshold. For arbitrary product noise, an effective $X$-state singular-value flow gives the teleportation usefulness, CJWR witness, and CHSH-nonlocal thresholds. If a local noise factor is pure, the entanglement threshold and all four steerability thresholds are zero. For generic full-rank mixed $X$ noise, finite-setting semidefinite programs give upper bounds on the unknown directional projective-measurement steerability thresholds.