Photon loss effects on light-mediated non-Gaussian entangled Bose-Einstein condensates projecting with different photon measurement outcomes

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.