Fully smooth one shot multipartite soft covering of quantum states without pairwise independence
Pranab Sen
TL;DR
This work overcomes the longstanding barrier that telescoping-based proofs of fully smooth multipartite convex split and soft covering require pairwise independence by introducing a novel machinery built from tilting, augmentation smoothing, and a fidelity-preserving flattening operation. The authors establish fully smooth one-shot multipartite convex split and smooth soft covering lemmas without relying on pairwise independence, and demonstrate powerful applications by deriving inner bounds for private classical communication over a quantum wiretap interference channel. Their framework hinges on connecting smooth Rényi-style divergences, projector smoothing, and flattening to control multipartite overlaps and privacy guarantees, providing a unified approach closer to simultaneous smoothing than to telescoping. Beyond the core results, the paper also presents a simpler convex-split proof and a fully smooth multipartite decoupling theorem, highlighting the broad applicability of the machinery to one-shot quantum network tasks. The findings offer significant implications for privacy-preserving communication in quantum networks and open avenues for further work on expander-walks, dimensional efficiency, and additional network information-theoretic problems.
Abstract
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 (arXiv:2410.17893) 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.