MOST BOSON QUANTUM STATES ARE ALMOST MAXIMALLY ENTANGLED

The geometric measure E of entanglement of an m qubit quantum state takes maximal possible value m. In previous work of Gross, Flammia, and Eisert, it was shown that E ≥ m − O(log m) with high probability as m → ∞. They show, 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 Boson quantum states (those that are actually available in current designs for quantum computers), the maximal possible geometric measure of entanglement is log2 m, opening the door to many computationally universal states. We further show the corresponding concentration result that E ≥ log2 m − O(log log m) with high probability as m → ∞. We extend these results also to m-mode n-bit Boson quantum states.