is given by$$X=\sqrt{\eta}I_{2},\qquad Y=(1-\eta)(2N_{B}+1)I_{2},
\label{eq:thermal-ch-Gaussian}$$ where $I_{2}$ is the $2\times2$ identity matrix. Our principal focus in this paper is on the thermal channel.

Teleportation simulation and reduction by teleportation {#sec:tp-simulation}
-------------------------------------------------------

Teleportation simulation of a channel [@BDSW96; @WPG07; @NFC09; @Mul12] is a key tool used to establish the upper bounds in , , , and . The basic idea behind this tool is that channels with sufficient symmetry can be simulated by the action of a teleportation protocol [@PhysRevLett.70.1895; @prl1998braunstein; @Werner01] on a resource state $\omega_{AB}$ shared between the sender $A$ and receiver $B$. More generally, a channel $\mathcal{N}_{A^{\prime}\rightarrow B}$ with input system $A^{\prime}$ and output system $B$ is defined to be teleportation simulable with associated resource state $\omega_{AB}$ if the following equality holds for all input states $\rho_{A^{\prime}}$:$$\mathcal{N}_{A^{\prime}\rightarrow B}(\rho_{A^{\prime}})=\mathcal{T}_{A^{\prime}AB}(\rho_{A^{\prime}}\otimes\omega_{AB}),\label{eq:TP-simulable-channel}$$ where $\mathcal{T}_{A^{\prime}AB}$ is a quantum channel consisting of local operations and classical communication between the sender, who has systems $A^{\prime}$ and $A$, and the receiver, who has system $B$ ($\mathcal{T}_{A^{\prime}AB}$ can also be considered a generalized teleportation protocol, as in [@Werner01]). The definition in was first given in [@WTB16arxiv], based on many earlier developments [@BDSW96; @Werner01; @WPG07; @NFC09; @Mul12]. The implication of channel simulation via teleportation is that the performance 