  Beam splitters are optical elements widely used in modern technological
applications to split the initial light beam into a required number of beams
and they play a very promising role for generating entangled optical states.
Here, a potential scheme is proposed to generate Bell coherent-states
superpositions through the action of a beam splitter when a Glauber coherent
state is injected on one input mode and vacuum state is incident on the other
one. Different quantifiers are used to measure the quantumness in the output
state such as concurrence entanglement, entropic quantum discord, quantum
coherence, geometric measure of quantum discord, local quantum uncertainty
(LQU) and local quantum Fisher information. Thereby, we derive their analytical
formulas and focus more on the behavior and bounds of each measure. Besides, we
have introduced the notion of "weak measurement-induced LQU" captured by weak
measurements as the generalization of normal LQU defined for standard
projective measurement, and we investigate the effect of the measurement
strength on the estimated phase enhancement if the generated Bell cat states
are the probe states in quantum metrology. Our results suggest that the
sensitivity of the interferometric phase estimation depends on how strongly one
perturbs the probe state and that a weak measurement does not necessarily
capture more quantumness in composite system.
