  The theory of quantum information processing for macroscopic qubits is based
on the fact that every macroscopic qubit has a conserved number of particles.
However, from an experimental point of view, every such qubit experiences
processes of decoherence that impact the possibilities for entanglement
generation between such qubits and use in quantum information processing
efficiently. One of the most prospective methods for generating entanglement
between distant atomic BECs is quantum nondemolition measurements. Here, we
study how the effects of photon measurement impact the entanglement when photon
loss decoherence is included. We employ the thermally entangled state
representation (TESR) and integral within the ordered operator(IWOP) approach
to obtain the accurate density matrix in a photon loss channel. We demonstrate
that varying outcomes of photon number measurements lead to the generation of
distinct entangled states, each exhibiting unique characteristics. We find that
using the Hofmann-Takeuchi and Duan-Giedke-Cirac-Zoller criterion provides
advantages in entanglement detection compared to the Wineland squeezing and EPR
steering criterion in such settings.
