A judicious application of the so-called pretty-good measurement in quantum coding considerably simplifies the derivation of one-shot achievability, sharpening many existing quantum information-processing protocols.
In this work, we prove a one-shot random coding bound for classical-quantum channel coding, a problem conjectured by Burnashev and Holevo in 1998. By choosing the optimal input distribution, the bound implies the optimal error exponent (i.e., the reliability function) of classical-quantum channels for rates above the critical rate, even in infinite-dimensional Hilbert spaces. Our result extends to various quantum packing-type problems, including classical communication over any fully quantum channel with or without entanglement-assistance, constant composition codes, and classical data compression with quantum side information via fixed-length or variable-length coding. Our technical ingredient is to establish an operator layer cake theorem - the directional derivative of an operator logarithm admits an integral representation of certain projections. This shows that a kind of pretty-good measurement is equivalent to a randomized Holevo-Helstrom measurement, which provides an operational explanation of why the pretty-good measurement is pretty good.
This paper introduces the induced divergence, a new quantum divergence measure that replaces the hypothesis testing divergence in position-based decoding, simplifying the analysis of quantum communication and state redistribution while yielding tighter achievability bounds. Derived from a parent quantum relative entropy, it retains key properties such as data processing inequality and Löwner monotonicity. Like the hypothesis testing divergence, it depends on a smoothing parameter and interpolates between the parent relative entropy (as the smoothing parameter approaches one) and the min-relative entropy (as it approaches zero), the latter holding when applied to the sandwiched Rényi relative entropy of order α∈[0,2]. This framework refines the position-based decoding lemma, extending its applicability to a broader class of states and improving decoding success probabilities. Two key applications are considered: classical communication over quantum channels, where the induced divergence improves lower bounds on the distillable communication rate, and quantum state redistribution, where it leads to sharper bounds on communication costs. These results provide new insights into fundamental single-shot quantum information protocols and enhance existing analytical techniques.
In this paper, we propose an elegant and simple one-shot achievability proof for classical-quantum channel coding. 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 classical-quantum channel is upper bounded by the minimum error of distinguishing the joint state between channel input and output against (M−1)-many products of its marginals. Our result provides the tightest achievable one-shot capacity for classical-quantum channel coding, and it improves the third-order coding rate in the asymptotic scenario. Moreover, the derived bound directly leads to asymptotic achievability results in the large deviation, small deviation, and moderate deviation regimes in a unified manner.The full version of the manuscript can be found at [arXiv:2208.02132] [1].