Statistics of Two Kinds of Entangled Quantum Many-body Systems

In this paper, we show two kinds of entangled many body systems with special statistic properties. Firstly, an entangled fermions system with a pairwise entanglement between every two particles in the lowest energy energy level obeys the fractional statistics. As a check, for particle number N=2, N=3 and N=4, considering that any two fermions in the lowest Landau level are entangled in a proper way, the Laughlin wave function can be derived. The results reveals the explicit entanglement pattern of the Laughlin states. Secondly, we noticed that both Bose-Einstein statistics and Fermi-Dirac distributions are derived from computing the partial function of a free quantum many body system in a certain ensemble without considering entanglement. We extend the computation of the partial function to an entangled quantum many body system without interaction, in this system we assume that every particle in energy level $\epsilon_i$ is entangled with a particle in the energy level $\epsilon_{i+1}$ ($i=1,3,5,...$) and also every particle in energy level $\epsilon_i+1$ is entangled with a particle in the energy level $\epsilon_{i}$ ($i=1,3,5,...$), which indicates that the two energy level have the same number of particles. In the entangled system, we find that the partial function will be changed. As a results, both the Bose-Einstein Statics and the Fermi-Dirac distributions will be modified at finite temperature.