Spin correlations and consequences of quantum-mechanical coherence

The difference in the properties of the spin correlation tensor for factorizable and nonfactorizable two-particle states is analyzed. The inequalities for linear combinations of the components of this tensor are obtained for the case of incoherent mixtures of factorizable two-particle spin states. They include the well known Bell inequalities and can be violated for coherent superpositions of two-particle spin states. The possibility to verify the consequences of the quantum-mechanical coherence is discussed using the angular correlations in the asymmetric (parity violating) decays of the pairs of spin-1/2 particles (muons, top-quarks or Λ-hyperons), the coherence arising either from the production dynamics or due to the effect of quantum statistics.