The emergence of typical entanglement in two-party random processes

Abstract

We investigate the entanglement within a system undergoing a random, local\nprocess. We find that there is initially a phase of very fast generation and\nspread of entanglement. At the end of this phase the entanglement is typically\nmaximal. In previous work we proved that the maximal entanglement is reached to\na fixed arbitrary accuracy within $O(N^3)$ steps, where $N$ is the total number\nof qubits. Here we provide a detailed and more pedagogical proof. We\ndemonstrate that one can use the so-called stabilizer gates to simulate this\nprocess efficiently on a classical computer. Furthermore, we discuss three ways\nof identifying the transition from the phase of rapid spread of entanglement to\nthe stationary phase: (i) the time when saturation of the maximal entanglement\nis achieved, (ii) the cut-off moment, when the entanglement probability\ndistribution is practically stationary, and (iii) the moment block entanglement\nscales exhibits volume scaling. We furthermore investigate the mixed state and\nmultipartite setting. Numerically we find that classical and quantum\ncorrelations appear to behave similarly and that there is a well-behaved\nphase-space flow of entanglement properties towards an equilibrium, We describe\nhow the emergence of typical entanglement can be used to create a much simpler\ntripartite entanglement description. The results form a bridge between certain\nabstract results concerning typical (also known as generic) entanglement\nrelative to an unbiased distribution on pure states and the more physical\npicture of distributions emerging from random local interactions.\n