  Out-of-time-ordered correlators (OTOCs) have been extensively used over the
last few years to study information scrambling and quantum chaos in many-body
systems. In this paper, we extend the formalism of the averaged bipartite OTOC
of Styliaris et al [Phys. Rev. Lett. 126, 030601 (2021)] to the case of open
quantum systems. The dynamics is no longer unitary but it is described by more
general quantum channels (trace preserving, completely positive maps). This
"open bipartite OTOC" can be treated in an exact analytical fashion and is
shown to amount to a distance between two quantum channels. Moreover, our
analytical form unveils competing entropic contributions from information
scrambling and environmental decoherence such that the latter can obfuscate the
former. To elucidate this subtle interplay we analytically study special
classes of quantum channels, namely, dephasing channels, entanglement-breaking
channels, and others. Finally, as a physical application we numerically study
dissipative many-body spin-chains and show how the competing entropic effects
can be used to differentiate between integrable and chaotic regimes.
