the Berry–Esseen theorem [@KS10; @S11], but many prior works have shown that first- and second-order terms like the above one serve as an excellent approximation for non-asymptotic capacities even for small $n$ [@polyanskiy10; @P10; @MW12; @P13; @TBR15]. The main new tool that we use to establish this result, beyond those used and introduced in [@WTB16], is a recent development in [@LMGA17] regarding teleportation simulation of single-mode phase-insensitive bosonic channels using finite-energy resource states. Figure \[fig:results\] plots this bound for several realistic values of the distance $L$ (related to transmissivity $\eta$) and thermal mean photon number $N_{B}$, and we point to Section \[sec:results\] for a more detailed discussion of these figures.

In the remainder of the paper, we argue how to arrive at the bound in . In what follows, we review the formalism of quantum Gaussian states and channels [@adesso14; @S17], and we also review information quantities needed, such as quantum relative entropy and relative entropy variance. We then review the critical tool of teleportation simulation of a quantum channel [@BDSW96; @WPG07; @NFC09; @Mul12] and how it can be used with [@WTB16 Eq. (4.34)] and ideas from [@LMGA17] in order to arrive at . We finally close with a summary 