[@BLTW16] can be expressed in terms of their first and second moments. For simplicity, let us suppose that $\rho$ and $\sigma$ are zero-mean quantum Gaussian states. Then Refs. [@PhysRevA.71.062320], [@PLOB15] established that$$D(\rho\Vert\sigma)=\log_{2}(Z^{\sigma}/Z^{\rho})/2-\operatorname{Tr}\{\Delta
V^{\rho}\}/4\ln2, \label{eq:rel-ent-G}$$ where $\Delta=G^{\rho}-G^{\sigma}$, and Ref. [@BLTW16] established that$$V(\rho\Vert\sigma)=\frac{1}{8\ln^{2}2}\left[  \operatorname{Tr}\{\Delta
V^{\rho}\Delta V^{\rho}\}+\operatorname{Tr}\{\Delta\Omega\Delta\Omega
\}\right]  . \label{eq:rel-ent-var-G}$$ In the above, we should note that our convention for normalization of covariance matrices is what leads to the different constant prefactors when compared to the expressions in [@PhysRevA.71.062320; @PLOB15; @BLTW16].

The hypothesis testing relative entropy is defined as [@BD10; @WR12]$$\begin{gathered}
D_{H}^{\varepsilon}(\rho\Vert\sigma)=\\
-\log_{2}\inf_{\Lambda}\{\operatorname{Tr}\{\Lambda\sigma\}:0\leq\Lambda\leq
I\wedge\operatorname{Tr}\{\Lambda\rho\}\geq1-\varepsilon\}\end{gathered}$$ By the reasoning in [@DPR15] and Appendix \[sec:AEP-hypo\], we have the following bound holding for faithful states $\rho$ and $\sigma$ such that $D(\rho\Vert\sigma),V(\rho\Vert\sigma),T(\rho\Vert\sigma)<\infty$ and $V(\rho\Vert\sigma)>0$:$$\begin{gathered}
D_{H}^{\varepsilon}(\rho^{\otimes n}\Vert\sigma^{\otimes n})\leq\\
nD(\rho\Vert\sigma)+\sqrt{nV(\rho\Vert\sigma)}\Phi^{-1}(\varepsilon)+O(\log
n), \label{eq:upper-bnd-hypo}$$ where [@li12; @TH12]$$T(\rho\Vert\sigma)=\sum_{x,y}\left\vert \left\langle \psi_{y}|\phi
_{x}\right\rangle \right\vert ^{2}\lambda_{x}\left\vert \log_{2}\left(
\lambda_{x}/\mu_{y}\right)  -D(\rho\Vert\sigma)\right\vert ^{3},$$ and $$\begin{aligned}
\Phi(a)  &  =\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{a} dx \ \exp\!\left(
\frac{-x^{2}}{2}\right) \\
\Phi^{-1}(\varepsilon)  &  =\sup\left\{  a\in\mathbb{R} \ |\ \Phi(a)
\leq\varepsilon\right\}  . \label{eq:Phi-inv}$$ We note here that the finiteness of $T(\rho\Vert\sigma)$ for finite-energy, faithful Gaussian states is essential to the main result of our paper. Inspecting the proof given in Appendix \[sec:AEP-hypo\], we see that the condition $T(\rho\Vert\sigma) < \infty$ allows us to invoke the Berry-Esseen theorem [@KS10; @S11], 