Signal-pump entanglement in quantum k-photon down-conversion

We study the quantum dynamics of the k-photon down-conversion process. We concentrate our attention on the entanglement between the pump and the signal modes. We show that the degree of the entanglement between the pump and the signal depends on the initial statistics of the light field in both modes. Moreover, the higher the nonlinear process the stronger the entanglement is. The entanglement between the modes is related to the marginal entropy in the pump (signal) mode; i.e., the larger the entanglement, the larger the entropy is. In the quantum k-photon down-conversion the signal mode at t>0 is not generally in a pure state which restricts applicability of the parametric approximation.