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High dimensional entangled states have attracted much more attentions, due totheir strong nonlocality and much powerful capability for quantum informationprocessing. By the methods presented in this paper, arbitrary forms entangledqudits including symmetric and asymmetric forms could be generated with theweak cross-Kerr nonlinearity. These schemes are heralded by the detections ofsingle-photon detectors. If all the detectors do not register any singlephotons, the generation is success with the probability $1/n^{M}$ determined bydimension $n$ and partite $M$. Furthermore, these schemes work well even withthe common photon number nonresolving detectors, therefore they are feasiblewith the current experimental technology.

Heralded generation of symmetric and asymmetric entangled qudits with weak cross-Kerr nonlinearity