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\begin{document}
\title{When does noise increase the quantum capacity? - Supplemental Material}
\author{Fernando G.S.L. Brand\~ao}
\affiliation{Departamento de F\'isica, Universidade Federal de Minas Gerais, Belo Horizonte, Caixa Postal 702, 30123-970, MG, Brazil}
\author{Jonathan Oppenheim}
\affiliation{Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, U.K.}
\author{Sergii Strelchuk}
\affiliation{Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, U.K.}

\maketitle

{\textbf{Appendix A: Calculation of The Coherent Information in the Strong Nonconvexity Protocol.}}

The action of the convex combination can be represented as the collection of
three channels acting on the input state with certain probabilities: the
private channel $\chn_\pdit$, the identity channel ${\cal I}$ (when the input
state goes through the erasure channel and erasure does not occur), and the
erasing channel ${\cal E}$ (when the input state goes through the erasure
channel and erasure occurs deterministically).
The coherent information for these 
9 possible situations, is presented in the table below:
\begin{table}[H]\caption{Channels acted and the corresponding coherent
information at the end of the protocol}
\label{T1}
\centering
\begin{tabular}{ |c|c| }
\hline Coherent information & Channels acted  \\
\hline $\kappa(1-\kappa)(1-p) \log d$ & ${\cal N}_{\gamma^{(d)}}\otimes {\cal
I}$  \\
\hline $\kappa(1-\kappa)(1-p) \log d$ & $ {\cal I}\otimes {\cal
N}_{\gamma^{(d)}}$  \\
\hline $0$                                 & $ {\cal N}_{\gamma^{(d)}}\otimes
{\cal N}_{\gamma^{(d)}}$\\
\hline $(1-\kappa)^2(1-p)^2 \log d$   & ${\cal I} \otimes {\cal I}$ \\
\hline $(1-\kappa)^2 p(1-p) \log d$        & ${\cal E} \otimes {\cal I}$ \\
\hline $-(1-\kappa)^2p(1-p) \log d$        & ${\cal I} \otimes {\cal E}$ \\
\hline $-(1-\kappa)^2p^2 \log d$        & ${\cal E} \otimes {\cal E}$ \\
\hline $-\kappa(1-\kappa)p\log d$      & ${\cal N}_{\gamma^{(d)}}\otimes {\cal
E}$\\
\hline $0$      & ${\cal E}\otimes {\cal N}_{\gamma^{(d)}}$\\
\hline  \end{tabular}\end{table}

Since which channel acted is flagged on Bob's site, as well as 
whether the input was erased in the case where the erasure channel acted, the coherent information
is just a sum of all the entries of the table, divided by $2$ since the protocol takes $2n$ uses of the channel.\\

{\textbf{Appendix B: Superactivation with the Large Class of Erasure Channels.}}

{\textbf{Proof of Lemma 1:}}
We will first examine the case when the shield was successfully transmitted to Bob (erasure does not take place). 
This case amounts to evaluating $I_{c}(A\rangle B)_{\widetilde\gamma^{(d)}_{uner}}$. 
The application of the Alicki-Fannes inequality for quantum conditional entropy~\cite{alicki_fannes} gives us:
\begin{align}
|&I_{c}(A\rangle B)_{\gamma^{(d)}_{uner}}-I_{c}(A\rangle B)_{\widetilde\gamma^{(d)}_{uner}}| \\
&=|S(A|B)_{\gamma^{(d)}_{uner}}-S(A|B)_{\widetilde\gamma^{(d)}_{uner}}|\\
&\le 4\epsilon\log d+2h(\epsilon).
\end{align}\\\\
Using Eqn. (7) in the Letter: $I_{c}(A\rangle B)_{\gamma^{(d)}_{uner}}=\log d$, gives
\begin{align}
 I_{c}(A\rangle B)_{\widetilde\gamma^{(d)}_{uner}}\ge(1-4\epsilon) \log d-2h(\epsilon).
\end{align} 
When erasure takes place, we similarly obtain the lower bound for $I_{c}(A\rangle B)_{{\widetilde\gamma}^{(d)}_{er}}$: 
\begin{align}
 I_{c}(A\rangle B)_{{\widetilde\gamma}^{(d)}_{er}}&\ge I_{c}(A\rangle B)_{{\gamma}^{(d)}_{er}}-4\epsilon \log d-2h(\epsilon)\\
&\ge-4\epsilon \log d-2h(\epsilon).
\end{align}
Recalling that, analogously to Eqn. (6) in the Letter, we have $I_c(A\rangle B)_{{\widetilde\gamma}^{(d)}} = (1-p)I_c(A\rangle B)_{{\widetilde\gamma}^{(d)}_{uner}}+ pI_c(A\rangle B)_{{\widetilde\gamma}^{(d)}_{er}}$, and substituting the expressions for the coherent information we obtain the final result: 
\begin{align}
{\cal Q}({\cal \widetilde{N}}_{\gamma^{(d)}}\otimes {\cal N}_e^p) &\ge I_c(A\rangle B)_{{\widetilde\gamma}^{(d)}} \\
&\ge (1-p-4\epsilon)\log d-2h(\epsilon).
\end{align}
 $\blacksquare$

\begin{figure}
\centering
\includegraphics[scale=0.65]{fig2.pdf}

\caption{The lower bound on quantum capacity $g(p,\epsilon)= (1-p-4\epsilon)\log d-2h(\epsilon)$ for $p\in [\frac{1}{2},1)$ and $\epsilon=\epsilon(p).$ }
\label{noisysa}
\end{figure}
As Fig. \ref{noisysa} demonstrates, for each channel ${\cal N}_e^p$ with the erasure probability $p\in [\frac{1}{2},1)$ there exists $\epsilon = \epsilon(p)$, and the pair $(p, \epsilon(p))$ satisfies $I_c(A\rangle B)_{{\widetilde\gamma}^{(d)}}>0$.

{\textbf{Appendix C: Superactivation With Depolarizing Channel.}}

\begin{figure}[h]
\centering
\includegraphics[scale=0.65]{fig3.pdf}
\caption{Superactivation using depolarizing channel with noisy resources.}
\label{sup_depol_with_noise}
\end{figure}
\bibliographystyle{apsrev}
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