A multi-sender decoupling theorem and simultaneous decoding for the quantum MAC

In this work, we prove a novel one-shot ‘multi-sender’ decoupling theorem generalising Dupuis' seminal single sender decoupling theorem. We start off with a multipartite quantum state, say on
$A_{1}A_{2}R$
, where
$A_{1}, A_{2}$
are treated as the two ‘sender’ systems and
$R$
is the reference system. We apply independent Haar random unitaries in tensor product on
$A_{1}$
and
$A_{2}$
and then send the resulting systems through a quantum channel. We want the channel output
$B$
to be almost in tensor with the untouched reference
$R$
. Our main result shows that this is indeed the case if suitable entropic conditions are met. An immediate application of our main result is to obtain a one-shot simultaneous decoder for sending quantum information over a
$k$
-sender entanglement unassisted quantum multiple access channel (QMAC). The rate region achieved by this decoder is the natural one-shot quantum analogue of the pentagonal classical rate region. Assuming a simultaneous smoothing conjecture, this one-shot rate region approaches the optimal rate region of Yard et al. [20] in the asymptotic iid limit. Our work is the first one to obtain a non-trivial simultaneous decoder for the QMAC with limited entanglement assistance in both one-shot and asymptotic iid settings; previous works used unlimited entanglement assistance.