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

2024-10-23

arXiv.org

AI Summary

This work addresses the long-standing limitation in quantum many-body systems where convex splitting (i.e., multipartite soft covering) traditionally relies on marginal pairwise independence. We establish the first fully smoothed one-shot multipartite soft covering theory that dispenses with this assumption. Methodologically, we introduce the novel “flattening” operation on quantum states—proving its fidelity invariance—and integrate it with tilting and augmentation smoothing, extended via telescoping techniques to achieve robust soft covering for non-independent marginal states. Our principal contributions are: (i) breaking the independence constraint inherent in conventional convex splitting theory; and (ii) deriving a natural one-shot inner bound for private classical communication over quantum wiretap–interference channels—a result unprecedented in both classical and quantum information theory.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

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.

Problem

• Develops smooth multipartite quantum covering without pairwise independence
• Extends convex split techniques for classical-quantum states
• Enables private classical communication over quantum wiretap channels

Innovation

• Leverages tilting and augmentation smoothing
• Uses flattening to preserve fidelity
• Enables non-pairwise independent results