system to a reservoir $R$ at zero temperature, and then ensure the quantum system does not reflect any particles. In this case the heat current equals $$\begin{aligned}
J^{\rm qb}_L = {\pi^2 \over 6h} N {k_{\rm B}}^2 T_L^2,
\label{Eq:Jqb}\end{aligned}$$ where $N$ is the number of transverse modes in the quantum system. We refer to this as the quantum bound (qb) on heat flow, because it depends on the quantum wave nature of the electrons; it depends on $N$, which is given by the cross-sectional area of the quantum system divided by $\lambda_{\rm F}^2$, where $\lambda_{\rm F}$ is the electron’s Fermi wavelength. As such $J_L^{\rm qb}$ is ill-defined within classical thermodynamics.

Uniquely defining temperature, heat and entropy {#Sect:Unique}
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![\[Fig:Unique\] To implement the procedure in Section \[Sect:Unique\], one starts with the circuit unconnected, as in (a), one then connects the circuit, as in (b). After a long time $t_{\rm expt}$, one disconnects the circuit, returning to (a). The circles are the quantum thermoelectrics, as in Fig. \[Fig:thermocouple\]. ](figure3.pdf){width="\columnwidth"}

Works on classical thermodynamics have shown that the definition of heat and entropy flows can be fraught with difficulties. For example, the rate of change of entropy cannot always be uniquely defined in classical 