A Simple and Tighter Derivation of Achievability for Classical Communication over Quantum Channels

Achievability in information theory refers to demonstrating a coding strategy that accomplishes a prescribed performance benchmark for the underlying task. In quantum information theory, the crafted Hayashi-Nagaoka operator inequality is an essential technique in proving a wealth of one-shot achievability bounds since it effectively resembles a union bound in various problems. In this work, we show that the pretty-good measurement naturally plays a role as the union bound as well. A judicious application of it considerably simplifies the derivation of one-shot achievability for classical-quantum (c-q) channel coding via an elegant three-line proof. The proposed analysis enjoys the following favorable features: (i) The established one-shot bound admits a closed-form expression as in the celebrated Holevo-Helstrom Theorem. Namely, the average error probability of sending $M$ messages through a c-q channel is upper bounded by the error of distinguishing the joint state between channel input and output against $(M-1)$-many products of its marginals. (ii) Our bound directly yields asymptotic results in the large deviation, small deviation, and moderate deviation regimes in a unified manner. (iii) The coefficients incurred in applying the Hayashi-Nagaoka operator inequality are no longer needed. Hence, the derived one-shot bound sharpens existing results that rely on the Hayashi-Nagaoka operator inequality. In particular, we obtain the tightest achievable $\epsilon$-one-shot capacity for c-q channel heretofore, and it improves the third-order coding rate in the asymptotic scenario. (iv) Our result holds for infinite-dimensional Hilbert space. (v) The proposed method applies to deriving one-shot bounds for data compression with quantum side information, entanglement-assisted classical communication over quantum channels, and various quantum network information-processing protocols

One-Shot Mutual Covering Lemma and Marton's Inner Bound with a Common Message

By developing one-shot mutual covering lemmas, we derive a one-shot achievability bound for broadcast with a common message which recovers Marton's inner bound (with three auxiliary random variables) in the i.i.d.~case. The encoder employed is deterministic. Relationship between the mutual covering lemma and a new type of channel resolvability problem is discussed.Comment: 6 pages; extended version of ISIT pape

A hypothesis testing approach for communication over entanglement assisted compound quantum channel

We study the problem of communication over a compound quantum channel in the presence of entanglement. Classically such channels are modeled as a collection of conditional probability distributions wherein neither the sender nor the receiver is aware of the channel being used for transmission, except for the fact that it belongs to this collection. We provide near optimal achievability and converse bounds for this problem in the one-shot quantum setting in terms of quantum hypothesis testing divergence. We also consider the case of informed sender, showing a one-shot achievability result that converges appropriately in the asymptotic and i.i.d. setting. Our achievability proof is similar in spirit to its classical counterpart. To arrive at our result, we use the technique of position-based decoding along with a new approach for constructing a union of two projectors, which can be of independent interest. We give another application of the union of projectors to the problem of testing composite quantum hypotheses.Comment: 21 pages, version 3. Added an application to the composite quantum hypothesis testing. Expanded introductio

The Likelihood Encoder for Lossy Compression

A likelihood encoder is studied in the context of lossy source compression. The analysis of the likelihood encoder is based on the soft-covering lemma. It is demonstrated that the use of a likelihood encoder together with the soft-covering lemma yields simple achievability proofs for classical source coding problems. The cases of the point-to-point rate-distortion function, the rate-distortion function with side information at the decoder (i.e. the Wyner-Ziv problem), and the multi-terminal source coding inner bound (i.e. the Berger-Tung problem) are examined in this paper. Furthermore, a non-asymptotic analysis is used for the point-to-point case to examine the upper bound on the excess distortion provided by this method. The likelihood encoder is also related to a recent alternative technique using properties of random binning