Entangled Brachistochrone: Minimum Time to reach Target Entangled State

We address the question: Given an arbitrary initial state and a general physical interaction what is the minimum time for reaching a target entangled state? We show that the minimum time is inversely proportional to the quantum mechanical uncertainty in the non-local Hamiltonian. We find that the presence of initial entanglement helps to minimize the waiting time. Furthermore, we find that in a bi-local rotating frame the entangling capability is actually a geometric quantity. We give an universal bound for the time average of entanglement rate for general quantum systems. The time average of entanglement rate does not depend on the particular Hamiltonian, rather on the fluctuation in the Hamiltonian. There can be infinite number of nonlocal Hamiltonians which may give same average entanglement rate. We also prove a composition law for minimum time when the system evolves under a composite Hamiltonian.