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IEEE Transactions on Information Theory

One-Shot Marton Inner Bound for Classical-Quantum Broadcast Channel

Jaikumar Radhakrishnan, Pranab Sen, Naqueeb Ahmad Warsi

Published Feb 18, 2016• 27 citations
Publisher

Abstract

We consider the problem of communication over a classical-quantum broadcast channel with one sender and two receivers. Generalizing the classical inner bounds shown by Marton and the recent quantum asymptotic version shown by Savov and Wilde, we obtain one-shot inner bounds in the quantum setting. Our bounds are stated in terms of hypothesis testing and one-shot max divergences. These results give a full justification of the claims of Savov and Wilde in the classical-quantum asymptotic iid setting; the techniques also yield similar bounds in the information spectrum setting. We obtain these results using a different analysis of the random codebook argument; our method yields a classical one-shot Marton bound with a common message and a classical one-shot mutual covering lemma based on rejection sampling.

DOI
10.1109/tit.2016.2531795

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We consider the problem of communication over a classical-quantum broadcast channel with one sender and two receivers. Generalizing the classical inner bounds shown by Marton and the recent quantum asymptotic version shown by Savov and Wilde, we obtain one-shot inner bounds in the quantum setting. Our bounds are stated in terms of hypothesis testing and one-shot max divergences. These results give a full justification of the claims of Savov and Wilde in the classical-quantum asymptotic iid setting; the techniques also yield similar bounds in the information spectrum setting. We obtain these results using a different analysis of the random codebook argument; our method yields a classical one-shot Marton bound with a common message and a classical one-shot mutual covering lemma based on rejection sampling.
We consider the problem of communication over a classical-quantum broadcast channel with one sender and two receivers. Generalizing the classical inner bounds shown by Marton and the recent quantum asymptotic version shown by Savov and Wilde, we obtain one-shot inner bounds in the quantum setting. Our bounds are stated in terms of smooth min and max Renyi divergences. We obtain these results using a different analysis of the random codebook argument and employ a new one-shot classical mutual covering argument based on rejection sampling. These results give a full justification of the claims of Savov and Wilde in the classical-quantum asymptotic iid setting; the techniques also yield similar bounds in the information spectrum setting.
We explore the classical communication over quantum channels with one sender and two receivers, or with two senders and one receiver, in both one-shot and asymptotic regimes. First, for the quantum broadcast channel (QBC) and the quantum multi-access channel (QMAC), we study the classical communication assisted by no-signalling and positive-partial-transpose-preserving codes, and obtain efficiently computable one-shot bounds to assess the performance of classical communication. Second, we consider the asymptotic communication capability of communication over the QBC and QMAC. We derive an efficiently computable strong converse bound for the capacity region, which behaves better than the previous semidefinite programming strong converse bound for point-to-point channels. Third, we obtain a converse bound on the one-shot capacity region based on the hypothesis testing divergence between the given channel and a certain class of subchannels. As applications, we analyze the communication performance for some basic network channels, including the classical broadcast channels and a specific class of quantum broadcast channels.
We consider the one-shot coding of multiple access channels one-time pad. Suppose that the senders and the receiver share an entangled state in the multi-party systems. Using the position-based coding, each sender encodes the classical messages on their systems, then they transmit their systems over the ideal quantum channels. The receiver applies a decoding operator to all the systems in order to figure out which message was transmitted with respect to the quantum joint typicality lemma. We provide the upper bound of the information leakage by the convex split lemma. Combining quantum max-information with quantum hypothesis testing mutual information, we give the inner bound of the achievable rate.
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.
We provide the first inner bounds for sending private classical information over a quantum multiple access channel. We do so by using three powerful information theoretic techniques: rate splitting, quantum simultaneous decoding for multiple access channels, and a novel smoothed distributed covering lemma for classical quantum channels. Our inner bounds are given in the one shot setting and accordingly the three techniques used are all very recent ones specifically designed to work in this setting. The last technique is new to this work and is our main technical advancement. For the asymptotic iid setting, our one shot inner bounds lead to the natural quantum analogue of the best classical inner bounds for this problem. A full version of this paper is accessible at [5].
We study the transmission of two independent confidential classical messages over a quantum broadcast channel, one for each receiver. Each message must remain secret from the other receiver, including when that receiver knows its own message. For classical inputs and quantum outputs, we establish the classical Marton-type inner bound under average reliability and conditional strong secrecy. The encoder selects pairs of codewords from independently generated codebooks using normalized likelihood weights. We prove reliability through a change-of-distribution argument. Our main technical result is a bipartite classical-quantum resolvability theorem that accounts for the dependence created by pair selection and establishes secrecy for the same encoder. We also obtain a multi-letter capacity characterization and extend it to arbitrary quantum inputs under secrecy against the other receiver, together with the Stinespring environment. We recover confidential capacity regions for deterministic classical and degraded classical-quantum channels, with an explicit evaluation for the classical Blackwell channel. For coherent isometric extensions of injective deterministic classical broadcast channels, we show that the confidential classical capacity region equals that of the corresponding classical channel. We then compare confidential classical communication with quantum transmission. For the coherent isometric extension of the Blackwell channel, we determine the unassisted quantum-capacity region and show that some achievable confidential classical rate pairs lie outside it. The Platypus channel provides another example of this separation.