Relationship between the field local quadrature and the quantum discord of a photon-added correlated channel under the influence of scattering and phase fluctuation noise

We study quantum correlations and discord in a bipartite continuous variable hybrid system formed by linear combinations of coherent states (mathinner {|{alpha }rangle }) and single photon-added coherent states of the form (mathinner {|{psi }rangle }_{text {dp(pa)}}= mathcal {N}/sqrt{2} (hat{a}^dagger mathinner {|{alpha }rangle }_amathinner {|{alpha }rangle }_b pm hat{b}^dagger mathinner {|{alpha }rangle }_amathinner {|{alpha }rangle }_b)). We stablish a relationship between the quantum discord with a local observable (the quadrature variance for one subsystem) under the influence of scattering and phase fluctuation noise. For the pure states the quantum correlations are characterized by means of measurement induced disturbance (MID) with simultaneous quadrature measurements. In a scenario where homodyne conditional measurements are available we show that the MID provides an easy way to select optimal phases to obtain information of the maximal correlations in the channels. The quantum correlations of these entangled states with channel losses are quantitatively characterized with the quantum discord (QD) with a displaced qubit projector. We observe that as scattering increases, QD decreases monotonically. At the same time for the state (mathinner {|{psi }rangle }_{text {dp}}), QD is more resistant to high phase fluctuations when the average photon number (n_0) is bigger than zero, but if phase fluctuations are low, QD is more resistant if (n_0=0). For the dp model with scattering, we obtain an analytical expression of the QD as a function of the observable quadrature variance in a local subsystem. This relation allows us to have a way to obtain the degree of QD in the channel by just measuring a local property observable such as the quadrature variance. For the other model this relation still exists but is explored numerically. This relation is an important result that allows to identify quantum processing capabilities in terms of just local observables.