Entanglement distribution and total entanglement in a fermion model with long-range interaction

Abstract

How is two-party entanglement (TPE) distributed in many-body systems? This is a fundamental issue because the total TPE between one party and all the other parties, ${\mathcal{C}}^{N}$, is upper bounded by the Coffman, Kundu, and Wootters monogamy inequality, from which ${\mathcal{C}}^{N}\ensuremath{\le}\sqrt{N\ensuremath{-}1}$ can be proved by the geometric inequality. Here we explore the total entanglement ${\mathcal{C}}^{\ensuremath{\infty}}$ and the associated total tangle ${\ensuremath{\tau}}^{\ensuremath{\infty}}$ in a $p$-wave free-fermion model with long-range interaction, showing that ${\mathcal{C}}^{\ensuremath{\infty}}\ensuremath{\sim}O(1)$ and ${\ensuremath{\tau}}^{\ensuremath{\infty}}$ may become vanishingly small with increasing long-range interaction. However, we always find ${\mathcal{C}}^{\ensuremath{\infty}}\ensuremath{\sim}2\ensuremath{\xi}{\ensuremath{\tau}}^{\ensuremath{\infty}}$, where $\ensuremath{\xi}$ is the truncation length of entanglement, beyond which the TPE quickly vanishes; hence ${\ensuremath{\tau}}^{\ensuremath{\infty}}\ensuremath{\sim}1/\ensuremath{\xi}$. This relation is a direct consequence of the exponential decay of the TPE induced by the long-range interaction. When $\ensuremath{\xi}$ is comparable to the system size $N$, this relation is reduced to the Koashi, Bu\ifmmode \check{z}\else \v{z}\fi{}ek, and Imoto (KBI) upper bound in the fully connected Lipkin-Meshkov-Glick model and Dicke model. Thus this relation can be regarded as a generalization of the KBI bound in quantum models, which may have intriguing applications in the more complicated many-body models.