Spin entanglement, decoherence and Bohm's EPR paradox

We obtain criteria for entanglement and the EPR paradox for spin-entangled particles and analyse the effects of decoherence caused by absorption and state purity errors. For a two qubit photonic state, entanglement can occur for all transmission efficiencies. In this case, the state preparation purity must be above a threshold value. However, Bohm's spin EPR paradox can be achieved only above a critical level of loss. We calculate a required efficiency of 58%, which appears achievable with current quantum optical technologies. For a macroscopic number of particles prepared in a correlated state, spin entanglement and the EPR paradox can be demonstrated using our criteria for efficiencies {\eta} > 1/3 and {\eta} > 2/3 respectively. This indicates a surprising insensitivity to loss decoherence, in a macroscopic system of ultra-cold atoms or photons.