Quantum Theory and Local Hidden Variable Theory: General Features and   Tests for EPR Steering
  In a previous paper tests for entanglement for two mode systems involving identical massive bosons were obtained. In the present paper we consider sufficiency tests for EPR steering in such systems. We find that spin squeezing in any spin component, a Bloch vector test, the Hillery-Zubairy planar spin variance test and squeezing in two mode quadratures all show that the quantum state is EPR steerable. We also find a generalisation of the Hillery-Zubairy planar spin variance test for EPR steering. The relation to previous correlation tests is discussed. This paper is based on a detailed classification of quantum states for bipartite systems. States for bipartite composite systems are categorised in quantum theory as either separable or entangled, but the states can also be divided differently into Bell local or Bell non-local states in terms of local hidden variable theory (LHVT). For the Bell local states there are three cases depending on whether both, one of or neither of the LHVT probabilities for each sub-system are also given by a quantum probability involving sub-system density operators. Cases where one or both are given by a quantum probability are known as local hidden states (LHS) and such states are non-steerable. The steerable states are the Bell local states where there is no LHS, or the Bell non-local states. The relationship between the quantum and hidden variable theory classification of states is discussed. 
Bryan J Dalton, Barry M Garraway and Margaret D Reid
quant-ph