Generalized Entanglement Entropies of Quantum Designs
Abstract
The entanglement properties of random quantum states or dynamics are important to the study of a broad spectrum of disciplines of physics, ranging from quantum information to high energy and manybody physics. This Letter investigates the interplay between the degrees of entanglement and randomness in pure states and unitary channels. We reveal strong connections between designs (distributions of states or unitaries that match certain moments of the uniform Haar measure) and generalized entropies (entropic functions that depend on certain powers of the density operator), by showing that Rényi entanglement entropies averaged over designs of the same order are almost maximal. This strengthens the celebrated Page's theorem. Moreover, we find that designs of an order that is logarithmic in the dimension maximize all Rényi entanglement entropies and so are completely random in terms of the entanglement spectrum. Our results relate the behaviors of Rényi entanglement entropies to the complexity of scrambling and quantum chaos in terms of the degree of randomness, and suggest a generalization of the fast scrambling conjecture.
 Publication:

Physical Review Letters
 Pub Date:
 March 2018
 DOI:
 10.1103/PhysRevLett.120.130502
 arXiv:
 arXiv:1709.04313
 Bibcode:
 2018PhRvL.120m0502L
 Keywords:

 Quantum Physics;
 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 6 pages. Concise version of arXiv:1703.08104. v4: published version with some minor additions and corrections