  Photon anti-bunching, measured via the Hanbury-Brown-Twiss experiment, is one
of the key signatures of quantum light and is tied to sub-Poissonian photon
number statistics. Recently, it has been reported that photon anti-bunching or
conditional sub-Poissonian photon number statistics can be obtained via
second-order interference of mutually incoherent weak lasers and heralding
based on photon counting. Here, we report theoretical analysis on the limits of
manipulating conditional photon statistics via interference of weak lasers. It
is shown that conditional photon number statistics can become super-Poissonian
in such a scheme. We, however, demonstrate explicitly that it cannot become
sub-Poissonian, i.e., photon anti-bunching cannot be obtained in such a scheme.
We point out that incorrect results can be obtained if one does not properly
account for seemingly negligible higher-order photon number expansions of the
coherent state.
