Hao-Chung Cheng

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.

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.

In the study of Markovian processes, one of the principal achievements is the equivalence between the Phi-Sobolev inequalities and an exponential decrease of the Phi-entropies. In this work, we develop a framework of Markov semigroups on matrix-valued functions and generalize the above equivalence to the exponential decay of matrix Phi-entropies. This result also specializes to spectral gap inequalities and modified logarithmic Sobolev inequalities in the random matrix setting. To establish the main result, we define a non-commutative generalization of the carre du champ operator, and prove a de Bruijn's identity for matrix-valued functions. The proposed Markov semigroups acting on matrix-valued functions have immediate applications in the characterization of the dynamical evolution of quantum ensembles. We consider two special cases of quantum unital channels, namely, the depolarizing channel and the phase-damping channel. In the former, since there exists a unique equilibrium state, we show that the matrix Phi-entropy of the resulting quantum ensemble decays exponentially as time goes on. Consequently, we obtain a stronger notion of monotonicity of the Holevo quantity - the Holevo quantity of the quantum ensemble decays exponentially in time and the convergence rate is determined by the modified log-Sobolev inequalities. However, in the latter, the matrix Phi-entropy of the quantum ensemble that undergoes the phase-damping Markovian evolution generally will not decay exponentially. This is because there are multiple equilibrium states for such a channel. Finally, we also consider examples of statistical mixing of Markov semigroups on matrix-valued functions. We can explicitly calculate the convergence rate of a Markovian jump process defined on Boolean hypercubes, and provide upper bounds of the mixing time on these types of examples.