  Numerous inequalities involving moments of integrated intensities and
revealing nonclassicality and entanglement in bipartite optical fields are
derived using the majorization theory, non-negative polynomials, the matrix
approach, as well as the Cauchy-Schwarz inequality. Different approaches for
deriving these inequalities are compared. Using the experimental photocount
histogram generated by a weak noisy twin beam monitored by a
photon-number-resolving iCCD camera the performance of the derived inequalities
is compared. A basic set of ten inequalities suitable for monitoring the
entanglement of a twin beam is suggested. Inequalities involving moments of
photocounts (photon numbers) as well as those containing directly the elements
of photocount (photon-number) distributions are also discussed as a tool for
revealing nonclassicality.
