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The effect of neutrino quantum decoherence has attracted a growing interest during the last 15 years. The most of already performed studies deal with searches of neutrino quantum decoherence in the terrestrial reactor and solar neutrino experiments. The forthcoming large volume neutrino detectors (e.g., JUNO, Hyper-Kamiokande and DUNE) will provide a new frontier in high-statistics measurements of astrophysical neutrino fluxes that will give, in particular, an opportunity to study the effect of neutrino quantum decoherence also in supernovae fluxes. In [1] we suggested a new mechanism of neutrino quantum decoherence in the supernovae engendered by the neutrino interaction with an external environment and showed that it becomes significant in the region where the collective neutrino oscillations occur. In [2] we studied the influence of the neutrino quantum decoherence of neutrino mass states on bipolar collective neutrino oscillations. In the present work we generalize our study [2] and present new analytical results on the interplay of the neutrino quantum decoherence and the synchronized neutrino collective oscillations. We also discuss the possibility to detect the effect of neutrino quantum decoherence in supernovae neutrino fluxes in the long baseline neutrino experiments. [1] K.Stankevich, A.Studenikin, Neutrino quantum decoherence engendered by neutrino radiative decay, Phys.Rev.D 101, 056004 (2020). [2] K.Stankevich, A.Studenikin, Collective neutrino oscillations accounting for neutrino quantum decoherence, PoS ICHEP2020, 216 (2021). This work was supported by the Russian Foundation for Basic Research under grant No. 20-52-53022-GFEN-a. The work of KS was also supported by the Russian Foundation for Basic Research under grant No. 20-32-90107 and by the “BASIS” Foundation No. 20-2-2-3-1.