Most Boson quantum states are almost maximally entangled

TL;DR

This paper demonstrates that Boson quantum states can have significantly lower entanglement than general states, making them more suitable for quantum computation, with entanglement measures scaling as logarithm of the number of particles.

Contribution

It establishes the maximal geometric measure of entanglement for Boson states as logarithmic in the number of particles, contrasting with previous results for general states.

Findings

Boson states have maximal entanglement of log2 m.

Entanglement concentrates around log2 m - O(log log m).

Results extend to m-mode n-bit Boson states.

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.