Photon number discrimination without a photon counter and reconstructing non-Gaussian states of light

We demonstrate a new approach to photon number discrimination, relying only on continuous variable homodyne measurements. We apply this technique to the reconstruction of the non-Gaussian one, two and three photon subtracted squeezed vacuum state. This new technique is remarkable as it seemingly contradicts the accepted idea that non-Gaussian quantum states with negative Wigner functions cannot be produced using only the Gaussian resources and techniques exploited here.