Inseparability criteria based on matrices of moments

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Inseparability criteria for continuous and discrete bipartite quantum states based on moments of annihilation and creation operators are studied by developing the idea of Shchukin-Vogel criterion [Phys. Rev. Lett. 95, 230502 (2005)]. If a state is separable, then the corresponding matrix of moments is separable too. Thus, we derive generalized criteria, based on the separability properties of the matrix of moments, are thus derived. In particular, a new criterion based on realignment of moments in the matrix is proposed as an analogue of the standard realignment criterion for density matrices. Other inseparability inequalities are obtained by applying positive maps to the matrix of moments. Usefulness of the Shchukin-Vogel criterion to describe bipartite-entanglement of more than two modes is demonstrated: We obtain some previously known three-mode inseparability criteria originally derived from the Cauchy-Schwarz inequality, and we introduce new ones.