  The couplings by change of measure are applied to establish log-Harnack
inequality(equivalently the entropy-cost estimate) for conditional
McKean-Vlasov SDEs and derive the quantitative conditional propagation of chaos
in relative entropy for mean field interacting particle system with common
noise. For the log-Harnack inequality, two different types of couplings will be
constructed for non-degenerate conditional McKean-Vlasov SDEs with
multiplicative noise. As to the quantitative conditional propagation of chaos
in relative entropy, the initial distribution of interacting particle system is
allowed to be singular with that of limit equation. The above results are also
extended to conditional distribution dependent stochastic Hamiltonian system.
