with the narrowest transmission distributions (the largest $ZT$ in linear-response) are automatically the best thermoelectrics.

At this point we mention that other works[@Nakpathomkun-Xu-Linke2010; @Leijnse2010; @Meden2013; @Hershfield2013] have studied efficiencies for various systems with finite width transmission functions, for which power outputs can be finite. In particular, Ref. \[\] considered a boxcar transmission function, which is the form of transmission function that we have shown can be made optimal [@2014w-prl].

Pendry’s quantum bound on heat-flow {#Sect:Pendry}
-----------------------------------

An essential ingredient in this work is Pendry’s upper bound [@Pendry1983] on the heat-flow through a quantum system between two reservoirs of fermions. He found this bound using a scattering theory of the type discussed in Section \[Sect:ScatteringTheory\] below. It is a concrete example of a general principle due to Bekenstein [@Bekenstein], and the same bound applies in the presence of thermoelectric effects [@2012w-2ndlaw]. The bound on the heat flow out of reservoir $L$ is achieved when all the electrons and holes arriving at the quantum system from reservoir $L$ escape into reservoir $R$ without impediment, while there is no back-flow of electrons or holes from reservoir $R$ to L. The easiest way to achieve this is to couple reservoir $L$ through the quantum 