  Bose-Einstein condensates with balanced gain and loss in a double-well
potential have been shown to exhibit PT-symmetric states. As proposed by
Kreibich et al [Phys. Rev. A 87, 051601(R) (2013)], in the mean-field limit the
dynamical behaviour of this system, especially that of the PT-symmetric states,
can be simulated by embedding it into a Hermitian four-well system with
time-dependent parameters. In this paper we go beyond the mean-field
approximation and investigate many-body effects in this system, which are in
lowest order described by the single-particle density matrix. The conditions
for PT symmetry in the single-particle density matrix cannot be completely
fulfilled by using pure initial states. Here we show that it is mathematically
possible to achieve exact PT symmetry in the four-well many-body system in the
sense of the dynamical behaviour of the single-particle density matrix. In
contrast to previous work, for this purpose, we use mixed initial states
fulfilling certain constraints and use them to calculate the dynamics.
