Quantum Theory and Local Hidden Variable Theory: General Features and Tests for EPR Steering and Bell Non-locality

Bryan J Dalton, Margaret D Reid

Published 2016-11-28Version 1

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. Bell states are where the probability P(a,b|A,B,c) for measured outcomes a, b on sub-system observables A, B for state preparation process c, is given by a local hidden variable theory (LHVT) form P(a,b|A,B,c)=Sum{h}P(h|c)P(a|A,c,h)P(b|B,c,h) (where preparation c results in a probability distribution P(h|c) for hidden variables h, P(a|A,c,h), P(b|B,c,h) are probabilities for measured outcome a,b on sub-system observable A,B when the hidden variables are h. Quantum states where P(a,b|A,B,c) is not given by a LHVT form are Bell non-local and lead to Bell inequality violation experiments. For the Bell local states there are three cases depending on whether both, one of or neither of the sub-system LHVT probabilities 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 tests for EPR steering and Bell non-locality in such systems. We find that spin squeezing in the any spin component, a weak correlation test, the Hillery-Zubairy spin variance test and a two mode quadrature squeezing test all show that the LHS model fails, and hence the quantum state is steerable. We also find a strong correlation test and a stronger version of the Hillery-Zubairy spin variance test also show that EPR steering occurs. In addition we find a new test for Bell non-locality that applies for spin operator measurements.