  The speed limits on entanglement are defined as the maximal rate at which
entanglement can be generated or degraded in a physical process. We derive the
speed limits on entanglement, using the relative entropy of entanglement and
trace-distance entanglement, for unitary as well as for arbitrary quantum
dynamics, where we assume that the dynamics of the closest separable state can
be approximately described by the closest separable dynamics of the actual
dynamics of the system. For unitary dynamics of isolated bipartite systems
which are described by pure states, the rate of entanglement production is
bounded by the product of fluctuations of the system's driving Hamiltonian and
the surprisal operator, with an additional term reflecting the time-dependent
nature of the closest separable state. Removing restrictions on the purity of
the input and on the unitarity of the evolution, the two terms in the bound get
suitably altered.
  Furthermore, we find a lower bound on the time required to generate or
degrade a certain amount of entanglement by arbitrary quantum dynamics. We
demonstrate the tightness of our speed limits on entanglement by considering
quantum processes of practical interest.
