  Strong nonlinearity at the single photon level represents a crucial enabling
tool for optical quantum technologies. Here we report on experimental
implementation of a strong Kerr nonlinearity by measurement-induced quantum
operations on weak quantum states of light. Our scheme coherently combines two
sequences of single photon addition and subtraction to induce a nonlinear phase
shift at the single photon level. We probe the induced nonlinearity with weak
coherent states and characterize the output non-Gaussian states with quantum
state tomography. The strong nonlinearity is clearly witnessed as a change of
sign of specific off-diagonal density matrix elements in Fock basis.
