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A general attenuator $Φ_{λ, σ}$ is a bosonic quantum channel that acts by combining the input with a fixed environment state $σ$ in a beam splitter of transmissivity $λ$. If $σ$ is a thermal state the resulting channel is a thermal attenuator, whose quantum capacity vanishes for $λ\leq 1/2$. We study the quantum capacity of these objects for generic $σ$, proving a number of unexpected results. Most notably, we show that for any arbitrary value of $λ>0$ there exists a suitable single-mode state $σ(λ)$ such that the quantum capacity of $Φ_{λ,σ(λ)}$ is larger than a universal constant $c>0$. Our result holds even when we fix an energy constraint at the input of the channel, and implies that quantum communication at a constant rate is possible even in the limit of arbitrarily low transmissivity, provided that the environment state is appropriately controlled. We also find examples of states $σ$ such that the quantum capacity of $Φ_{λ,σ}$ is not monotonic in $λ$. These findings may have implications for the study of communication lines running across integrated optical circuits, of which general attenuators provide natural models.