Entanglement of a bipartite multiparticle mesoscopic superposition

Abstract

Entanglement of an open bipartite system is investigated. In particular, we calculate the entanglement of a subsystem $A$ with $k$ particles in an $N$ atoms of two-level system $(k\ensuremath{\leqslant}N∕2)$ initially prepared in a mesoscopic superposition $\ensuremath{\mid}\ensuremath{\psi}⟩=[{\ensuremath{\mid}{\ensuremath{\phi}}_{1}({\ensuremath{\theta}}_{0})⟩}^{\ensuremath{\bigotimes}N}+{\ensuremath{\mid}0⟩}^{\ensuremath{\bigotimes}N}]$, with $\ensuremath{\mid}{\ensuremath{\phi}}_{1}({\ensuremath{\theta}}_{0})⟩=\mathrm{cos}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\theta}}_{0}\ensuremath{\mid}0⟩+\mathrm{sin}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\theta}}_{0}\ensuremath{\mid}1⟩$, subject to the time evolution described by a dephasing channel. For an arbitrary number of particles $N$, numerical results are given for the full time evolution up to ten particles and $1\ensuremath{\leqslant}k\ensuremath{\leqslant}5$. Analytical results are obtained for short times and asymptotic time regimes. Entanglement is maximum when ${\ensuremath{\theta}}_{0}=\ensuremath{\pi}∕2$ [Greenberger-Horne-Zeilinger (GHZ) state], independently of the number of particles in each partition. As $N$ gets larger, entanglement tends to $1∕2$, and becomes independent of ${\ensuremath{\theta}}_{0}$. The ``thermodynamic'' regime ($N\ensuremath{\rightarrow}\ensuremath{\infty}$ with $fN$ finite, where $f=k∕N$) is reached already for $N=1000$ particles. In the regime of $k⪡N$ and $N\ensuremath{\rightarrow}\ensuremath{\infty}$, the entanglement has its maximum value at ${\ensuremath{\theta}}_{0}=\ensuremath{\pi}∕2$ and for ${\ensuremath{\theta}}_{0}=0,\ensuremath{\pi}$ its minimum value. The time evolution reduces the maximum value of entanglement, showing that the subsystem $A$ loses coherence to the environment. Finally, we point out the differences between the entanglement rate obtained here from that one in Phys. Rev. Lett. 89, 210402 (2002), where only one $N$-particle subsystem is used.