resource state, and our Appendix \[sec:finiteness\] argues how $T(\omega_{AB}\Vert\sigma_{AB})$ is finite for finite-energy Gaussian states. Thus, the Berry–Esseen theorem can be invoked as shown in Appendix \[sec:AEP-hypo\] and so applies. On the other hand, for the derivation of , the ideal infinite-energy Choi state of the channel is used as the resource state, but it is not known if $T(\omega_{AB}\Vert
\sigma_{AB})$ is finite in such a scenario. Hence, unless this is proven, we cannot invoke . Therefore, other techniques, such as the Chebyshev inequality, were used in [@WTB16] to arrive at .

Methods
=======

Given the background reviewed above, we are now in a position to discuss the main contribution of our paper. We modify the finite-energy teleportation simulation approach of [@LMGA17] in the following way: Given a thermal channel with $\tau=\eta$ and $y=(1-\eta)(2N_{B}+1)$, we find a finite-energy, two-mode Gaussian state in standard form such that

1.  it satisfies ,

2.  its smaller symplectic eigenvalue is just larger than one, and

3.  its suboptimal relative entropy of entanglement is equal to the suboptimal relative entropy of entanglement of the Choi state of the channel, the latter of which is given by .

Any resource state that simulates the channel should 