# Quantum Theory and Local Hidden Variable Theory: General Features and Tests for EPR Steering. (arXiv:1611.09101v4 [quant-ph] UPDATED)

Quantum states for bipartite composite systems are categorised as either

separable or entangled, but the states can also be divided differently into

Bell local or Bell non-local states. This paper presents a detailed

classification of quantum states for bipartite systems and describes the

interrelationships between the various types. For the Bell local states there

are three cases depending on whether both, one of or neither of the local

hidden variable theory 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. 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 the any spin

component, a Bloch vector test, the Hillery-Zubairy planar spin variance test

and a two mode quadrature squeezing test all show that the LHS model fails, and

hence 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.