Exact convergence times for generation of random bipartite entanglement

We calculate exact convergence times to reach random bipartite entanglement for various random protocols. The eigenproblem of a Markovian chain governing the process is mapped to a spin chain, thereby obtaining exact expression for the gap of the Markov chain for any number of qubits. For protocols coupling nearest-neighbor qubits and a controlled-NOT (CNOT) gate the mapping goes to the $XYZ$ model while for a U(4) gate it goes to an integrable $XY$ model. For coupling between a random pair of qubits the mapping is to an integrable Lipkin-Meshkov-Glick model. In all cases the gap scales inversely with the number of qubits, thereby improving on a recent bound [Phys. Rev. Lett. 98, 130502 (2007)].