  We provide the first inner bounds for sending private classical information
over a quantum multiple access channel. We do so by using three powerful
information theoretic techniques: rate splitting, quantum simultaneous decoding
for multiple access channels, and a novel smoothed distributed covering lemma
for classical quantum channels. Our inner bounds are given in the one shot
setting and accordingly the three techniques used are all very recent ones
specifically designed to work in this setting. The last technique is new to
this work and is our main technical advancement. For the asymptotic iid
setting, our one shot inner bounds lead to the natural quantum analogue of the
best classical inner bounds for this problem.
