Compatibility and separability for classical and quantum entanglement

We study the concepts of compatibility and separability and their implications for quantum and classical systems. These concepts are illustrated on a macroscopic model for the singlet state of a quantum system of two entangled spin 1/2 with a parameter reflecting indeterminism in the measurement procedure. By varying this parameter we describe situations from quantum, intermediate to classical and study which tests are compatible or separated. We prove that for classical deterministic systems the concepts of separability and compatibility coincide, but for quantum systems and intermediate systems these concepts are generally different. More specifically, equal physical constraints in the model, which for classical deterministic measurements lead to non-separability and non-compatibility of the measurements, give rise to compatible measurements that are non-separated whenever indeterminism is introduced. As a consequence, compatible measurements that are non-separated, and hence violate Bell's inequalities, can only be realized in the presence of indeterminism.