Fully smooth one shot multipartite soft covering of quantum states without pairwise independence

We provide a powerful machinery to prove fully smooth one shot multipartite covering, aka convex split, type results for quantum states. In the important case of smooth multipartite convex split for classical quantum states, aka smooth multipartite soft covering, our machinery works even when certain marginals of these states do not satisfy pairwise independence. The recent paper of Sen gave the first proof of fully smooth multipartite convex split by simplifying and extending a technique called telescoping, developed originally for convex split by (arXiv:2304.12056). However, that work as well as all earlier works on convex split assumed pairwise or even more independence amongst suitable marginals of the quantum states. We develop our machinery by leveraging known results from (arXiv:1806.07278) involving tilting and augmentation smoothing of quantum states, combined with a novel observation that a natural quantum operation `flattening' quantum states actually preserves the fidelity. This machinery is powerful enough to lead to non pairwise independent results as mentioned above. As an application of our soft covering lemma without pairwise independence, we prove the `natural' one shot inner bounds for sending private classical information over a quantum wiretap interference channel, even when the classical encoders at the input lose pairwise independence in their encoding strategies to a certain extent. This result was unknown earlier even in the classical setting.