Most Boson quantum states are almost maximally entangled

Abstract

The geometric measure of entanglement $E$ of an $m$ qubit quantum state takes maximal possible value $m$. In previous work of Gross, Flammia, and Eisert, it was shown that $E\ge m-O(\log m)$ with high probability as $m\to \infty$. They showed, as a consequence, that the vast majority of states are too entangled to be computationally useful. In this paper, we show that for $m$ qubit {\em Boson} quantum states (those that are actually available in current designs for quantum computers), the maximal possible geometric measure of entanglement is ${\log }_{2}m$, opening the door to many computationally universal states. We further show the corresponding concentration result that $E\ge {\log }_{2}m-O(\log \log m)$ with high probability as $m\to \infty$. We extend these results also to $m$-mode $n$-bit Boson quantum states.