  The concept of entangled quantum states is considered in the context of
systems of identical particles, based on the requirement that in order to
represent physical states both for the overall system and the sub-systems which
may be entangled, the density operators must satisfy the symmetrisation
principle and global and local super-selection rules that prohibit states in
which there are coherences between differing particle numbers. These
requirements and their justification are fully discussed. In the second
quantisation approach used, both the system and the sub-systems are modes (or
sets of modes) rather than particles, particles being associated with different
occupancies of the modes. The definition of entangled states is based on first
defining the non-entangled states - after specifying which modes constitute the
sub-systems. This paper mainly focuses on two mode entanglement for massive
bosons. Several inequalities involving variances and mean values of operators
involving mode annihilation, creation operators have been proposed as tests for
two mode entangled states, including the inequalities that define spin
squeezing. Spin squeezing criteria in two mode systems are examined, and spin
squeezing is best considered for principle spin operator components where the
covariance matrix is diagonal. It is shown that the presence of spin squeezing
in at least one of the spin components requires entanglement of the relevant
pair of modes. Several of the other proposed tests for entanglement, including
ones based on the sum of the variances for two spin components are considered.
All of the tests are still valid when the present concept of entanglement based
on the symmetrisation and super-selection rule criteria is applied, but further
tests have been obtained here. Sometimes the new tests are satisfied whilst
than those obtained in other papers are not.
