Most boson quantum states are almost maximally entangled
Shmuel Friedland, Todd Kemp · Proceedings of the American Mathematical Society · 2018
The geometric measure of entanglement E E of an m m qubit quantum state has maximum value bounded above by m m . In previous work of Gross, Flammia, and Eisert, it was shown that E ≥ m − O ( log m ) E \ge m-O(\log m) with high probability as m → ∞ 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 m qubit Boson quantum states, the maximal possible geometric measure of entanglement is bounded above by log 2 m \log _2\! m , opening the door to many computationally universal states. We further show the corresponding concentration result that E ≥ log 2 m − O ( log log m ) E \ge \log _2\! m - O(\log \log m) with high probability as m → ∞ m\to \infty . We extend these results also to m m -mode n n -bit Boson quantum states.