  Recent experimental advances probing coherent phonon and electron transport
in nanoscale devices at contact have motivated theoretical channel-based
analyses of conduction based on the nonequilibrium Green's function formalism.
The transmission through each channel has been known to be bounded above by
unity, yet actual transmissions in typical systems often fall far below these
limits. Building upon recently derived radiative heat transfer limits and a
unified formalism characterizing heat transport for arbitrary bosonic systems
in the linear regime, we propose new bounds on conductive heat transfer. In
particular, we demonstrate that our limits are typically far tighter than the
Landauer limits per channel and are close to actual transmission eigenvalues by
examining a model of phonon conduction in a 1-dimensional chain. Our limits
have ramifications for designing molecular junctions to optimize conduction.
