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Separability criteria are typically of the necessary, but not sufficient,variety, in that satisfying some separability criterion, such as positivity ofeigenvalues under partial transpose, does not strictly imply separability.Certifying separability amounts to proving the existence of a decomposition ofa target mixed state into some convex combination of separable states;determining the existence of such a decomposition is "hard." We show that it iseffective to ask, instead, if the target mixed state "fits" some preconstructedseparable form, in that one can generate a sufficient separability criterionrelevant to all target states in some family by ensuring enough degrees offreedom in the preconstructed separable form. We demonstrate this technique byinducing a sufficient criterion for "diagonally symmetric" states of N qubits.A sufficient separability criterion opens the door to study precisely howentanglement is (not) formed; we use ours to prove that, counterintuitively,entanglement is not generated in idealized Dicke model superradiance despiteits exemplification of many-body effects. We introduce a quantification of theextent to which a given preconstructed parametrization comprises the set of allseparable states; for "diagonally symmetric" states our preconstruction isshown to be fully complete. This implies that our criterion is necessary inaddition to sufficient, among other ramifications which we explore.

Certifying Separability in Symmetric Mixed States, and Superradiance