  We show that intrinsic fluctuations in system control parameters impose
limits on the ability of two-qubit (exchange) Hamiltonians to generate
entanglement starting from mixed initial states. We find three classes for
Gaussian and Laplacian fluctuations. For the Ising and XYZ models there are
qualitatively distinct sharp entanglement-generation transitions, while the
class of Heisenberg, XY, and XXZ Hamiltonians is capable of generating
entanglement for any finite noise level. Our findings imply that exchange
Hamiltonians are surprisingly robust in their ability to generate entanglement
in the presence of noise, thus potentially reducing the need for quantum error
correction.
