Dualistic operational characterization of device-dependent correlation sets via convex analysis in the $(2,m,2)$ Bell scenario

Abstract

We analyze device-dependent correlation sets generated by fixed local dichotomic measurements for two-qubit systems in the $(2,m,2)$ Bell scenario. We consider three fundamental state spaces for the composite system: the separable state space, the standard quantum state space, and the maximal tensor-product state space, which contains beyond-quantum states compatible with local quantum measurements. We formulate the corresponding correlation sets for general fixed dichotomic measurements and, in the traceless case, derive particularly simple explicit formulae for their support and gauge functions. These functions furnish dual operational characterizations of the three correlation sets: the support functions give optimal witnesses for entanglement and beyond-quantum states, whereas the gauge functions quantify the robustness of these detections against depolarizing noise. We further derive convex-hull representations that elucidate the extremal structures of the correlation sets and the physical states realizing them, showing in particular that extremal quantum correlations are realized by maximally entangled states. The fundamental limits of these dual operational tasks are governed solely by the smaller of the numbers of linearly independent measurement directions available to Alice and Bob. When both parties have three linearly independent measurement directions, our entanglement criterion detects Werner states up to the optimal PPT threshold $p_{\mathrm{crit}}=2/3$. For beyond-quantum-state detection, a nontrivial separation from the quantum set occurs only under the same measurement condition; in that case, the same optimal noise threshold is attained for an extremal state in the maximal tensor-product state space.