Almost Surely Convergence of the Quantum Entropy of Random Graph States and the Area Law

It is known that the ensemble of random pure quantum states, constructed by bipartite maximally entangled states and random unitary matrices generated according to the Haar measure, exhibits an average entanglement entropy that obeys an area law. Our goal is to explore the entanglement entropy between the two subsystems in more detail. By employing techniques from Weingarten calculus and flow problems, we derive inequalities related to permutations. These inequalities lead to the conclusion that entanglement entropy almost surely follows an area law. We assert that these results persist even when replacing the Haar unitary random matrix with the Gaussian unitary ensemble. Finally, we illustrate our main results through two concrete examples.