Plenary 3: f-Divergences & Quantum Hypothesis Testing / Quantum Channel Coding with Few Code Lengths

Abstract: Divergences lie at the core of information-theoretic applications. A recently introduced family of f-divergences, defined via an integral representation, has exhibited remarkable properties --- for instance, for the study of contraction coefficients. However, many familiar properties of their classical analogous have remained elusive. In this work, we develop alternative representations of the quantum f-divergences by leveraging the recently established quantum layer-cake theorem. These new formulations enable us to establish several key properties, including monotonicity and connections to other divergences. As our main application, we show how these representations unify and streamline various proofs in quantum hypothesis testing, yielding tighter achievability bounds through conceptually simple arguments that apply across different error regimes.

Abstract: A central problem in quantum channel coding is determining the code length needed to achieve a given error tolerance ε. We improve the previous second-order bound with quadratic dependence O(1/ε^2) to a logarithmic dependence O(log 1/ε) for small ε. This is achieved by proving a one-shot random coding bound for classical--quantum channel coding, a conjecture of Burnashev and Holevo (1998). Optimizing the input distribution, our result yields the optimal error exponent for classical--quantum channels at rates above the critical rate, even in infinite-dimensional Hilbert spaces. Furthermore, these results apply to classical communication over general quantum channels, with or without entanglement assistance. A key technical tool is our new operator layer cake theorem. This theorem shows that a form of the pretty-good measurement is equivalent to a randomized Holevo–Helstrom measurement, providing an operational explanation of why the pretty-good measurement is pretty good.