Alice and Bob communicate at a rate equal to the accessible information in the limit of infinite message size $N$.

There is no known general method for calculating the accessible information exactly. Here we will make use of the upper and lower bounds found by Fuchs and Caves [@PhysRevLett.73.3047]. The lower bound, hereby referred to as the Fuch’s lower bound, is $$\begin{gathered}
I_{\rm lower}=\Tr\Big\{p_0\rho^{(0)}\log_2\left[ \mathcal{L}_{\bar\rho}(\rho^{(0)}) \right]\\
 + p_1\rho^{(1)}\log_2\left[ \mathcal{L}_{\bar\rho}(\rho^{(1)})\right] \Big\}
\label{eqn_fuchslower}\end{gathered}$$ where $\mathcal{L}$ is the lowering superoperator given by $$\begin{gathered}
\mathcal{L}_{\bar\rho}(\Delta)= \sum_{\{j,k|\lambda_j+\lambda_k\neq 0\}} \bigg[ \frac{2}{\lambda_j(p_1)+\lambda_k(p_1)} \\ \times \bra{\psi_j(p_1)}\Delta\ket{\psi_j(p_1)}
 \ket{\psi_j(p_1)}\bra{\psi_k(p_1)}\bigg] \ ,\end{gathered}$$ and where $\Delta=\rho^{(1)}-\rho^{(0)}$. $\lambda_i(p_1)$ and $\ket{\psi_i(p_1)}$ are the eigenvalues and eigenvectors of $\bar\rho=(1-p_1)\rho^{(0)}+p_1\rho^{(1)}$. The Fuchs upper bound $I_{\rm upper}$ is found by numerically solving the differential equation, $$\begin{gathered}
\label{eqn_fuchsupper}
\frac{{\mathop{}\!\mathrm{d}}^2 I_{\rm upper}(p_1)}{{\mathop{}\!\mathrm{d}}p_1^2}=
 \sum_{\{ j,k|\lambda_j+\lambda_k\neq 0 \}} \bigg[ -\frac{2}{\lambda_j(p_1)+\lambda_k(p_1)} \\
 \times |\bra{\psi_j(p_1)}\Delta\ket{\psi_k(p_1)}|^2 \bigg]\end{gathered}$$ subject to $$I_{\rm upper}(0)=I_{\rm upper}(1)=0.$$

The other figure of merit we consider is the Holevo information [@holevo1973bounds]. It is given by $$\begin{gathered}
\label{eqn_holevo}
\chi (\rho^{(0)}, \rho^{(1)}) = S\left(\sum_{x=0}^1 p_x \rho^{(x)}\right) - \sum_{x=0}^1 p_x S(\rho^{(x)})\end{gathered}$$ where $S(\rho)$ is the von Neumann entropy of the quantum state $\rho$. The Holevo information is the maximum communication rate Bob can obtain, provided he stores all of the $N$ states and then 