Matrix Chernoff concentration bounds for multipartite soft covering and expander walks
Pranab Sen
TL;DR
This work establishes Chernoff-type exponential concentration bounds for classical-quantum soft covering in two new regimes: fully smooth multipartite coverage and unipartite coverage via expander-walk sampling. It combines a fully smooth expectation bound (via Sen:telescoping) with a McDiarmid-style concentration step, and introduces bounded excision to handle expander walks, yielding dimension-free tail bounds in trace distance that depend on smooth Rényi divergences. The results include a multipartite concentration bound that extends and strengthens prior Ahlswede–Winter results and an expander-walk bound that improves upon previous matrix Chernoff bounds by avoiding explicit Hilbert-space dimension factors. These bounds enable dimension-independent inner bounds for private classical communication over quantum wiretap channels with many non-interacting eavesdroppers and have potential applications in multiterminal quantum information problems. The techniques, especially bounded excision, offer a new tool for concentration phenomena in settings where samples arise from expander graphs rather than independent draws.
Abstract
We prove Chernoff style exponential concentration bounds for classical quantum soft covering generalising previous works which gave bounds only in expectation. Our first result is an exponential concentration bound for fully smooth multipartite classical quantum soft covering, extending Ahlswede-Winter's seminal result in several important directions. Next, we prove a new exponential concentration result for smooth unipartite classical quantum soft covering when the samples are taken via a random walk on an expander graph. The resulting expander matrix Chernoff bound complements the results of Garg, Lee, Song and Srivastava in important ways. We prove our new expander matrix Chernoff bound by generalising McDiarmid's method of bounded differences for functions of independent random variables to a new method of bounded excision for functions of expander walks. This new technical tool should be of independent interest. A notable feature of our new concentration bounds is that they have no explicit Hilbert space dimension factor. This is because our bounds are stated in terms of the trace distance of the sample averaged quantum state to the `ideal' quantum state. Our bounds are sensitive to certain smooth Renyi max divergences, giving a clear handle on the number of samples required to achieve a target trace distance. Using these novel features, we prove new one shot inner bounds for sending private classical information over different kinds of quantum wiretap channels with many non-interacting eavesdroppers that are independent of the Hilbert space dimensions of the eavesdroppers. Such powerful results were unknown earlier even in the fully classical setting.