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arXiv:2504.04067

Matrix Chernoff concentration bounds for multipartite soft covering and expander walks

Sen, Pranab

Published Apr 5, 2025• 0 citations

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.

DOI: 10.48550/arxiv.2504.04067

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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.

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.

In this work we study the problem of communication over the quantum wiretap channel. For this channel there are three parties Alice (sender), Bob (legitimate receiver) and Eve (eavesdropper). We obtain upper and lower bounds on the amount of information Alice can communicate to Bob such that Eve gets to know as little information as possible about the transmitted messages. Our bounds are in terms of quantum hypothesis testing divergence and smooth max quantum relative entropy. To obtain our result we prove a one-shot version of the quantum covering lemma along with operator Chernoff bound for non-square matrices.

How well can we approximate a classical-quantum channel output state by using a random codebook with a certain size? In this work, we study the quantum soft covering problem and establish exponential achievability and strong converse bounds on the expected trace distance between the codebook-induced state and the true state. When using independent and identically distributed random codebook or constant composition random codebook with a rate above the quantum mutual information I(X : B)ρ, we prove that the trace distances decay exponentially with error exponents expressed by the sandwiched Rényi and Augustin information. For a rate below I(X : B)ρ, we show that both the trace distances converge to 1 exponentially fast. The full manuscript can be found at [1].

How well can we approximate a quantum channel output state using a random codebook with a certain size? In this work, we study the quantum soft covering problem. Namely, we use a random codebook with codewords independently sampled from a prior distribution and send it through a classical-quantum channel to approximate the target state. When using a random codebook sampled from an independent and identically distributed prior with a rate above the quantum mutual information, we show that the expected trace distance between the codebook-induced state and the target state decays with exponent given by the sandwiched Rényi information. On the other hand, when the rate of the codebook size is below the quantum mutual information, the trace distance converges to one exponentially fast. We obtain similar results when using a random constant composition codebook, whereas the sandwiched Augustin information expresses the error exponent. In addition to the above large deviation analysis, our results also hold in the moderate deviation regime. That is, we show that even when the rate of the codebook size approaches the quantum mutual information moderately quickly, the trace distance still vanishes asymptotically.

How well can we approximate a quantum channel output state using a random codebook with a certain size? In this work, we study the quantum soft covering problem, which uses a pairwise-independent random codebook to approximate the target output state of a quantum channel. We establish a one-shot error exponent bound and a one-shot strong converse bound on the approximation error measured in terms of the expected trace distance between the codebook-induced state and the true channel output state. When using independent and identically-distributed random codebook with a rate above the quantum mutual informationI(X : B)ρ, we prove that the error decays exponentially with its error exponent expressed by the sandwiched Rényi information. On the other hand, when the rate of the codebook size is below the quantum mutual information, the error converges to one exponentially fast. Similar results are obtained using a random constant composition codebook, whereas the sandwiched Augustin information gives the error exponent. In addition to the above large deviation analysis, our results also hold in the moderate deviation regime.

We introduce an improved one-shot characterisation of randomness extraction against quantum side information (privacy amplification), strengthening known one-shot bounds and providing a unified derivation of the tightest known asymptotic constraints. Our main tool is a new class of smooth conditional entropies defined by lifting classical smooth divergences through measurements. A key role is played by the measured smooth Rényi relative entropy of order 2, which we show to admit an equivalent variational form: it can be understood as allowing for smoothing over not only states, but also non-positive Hermitian operators. Building on this, we establish a tightened leftover hash lemma, significantly improving over all known smooth min-entropy bounds on extractable randomness and recovering the sharpest classical achievability results. We extend these methods to decoupling, the coherent analogue of privacy amplification, obtaining a corresponding improved one-shot bound. Relaxing our smooth entropy bounds leads to one-shot achievability results in terms of measured Rényi divergences, tightening the bounds of [Dupuis, arXiv:2105.05342] and recovering state-of-the-art asymptotic i.i.d. error exponents. We show an approximate optimality of our results by giving a matching one-shot converse bound up to additive logarithmic terms. This yields an optimal second-order asymptotic expansion of privacy amplification under trace distance, establishing a significantly tighter one-shot achievability result than previously shown in [Shen et al., arXiv:2202.11590] and proving its optimality for all hash functions.