Hao-Chung Cheng

Concavity of the Auxiliary Function for Classical-Quantum Channels

The auxiliary function of a classical channel appears in two fundamental quantities, the random coding exponent and the sphere-packing exponent, which yield upper and lower bounds on the error probability of decoding, respectively. A crucial property of the auxiliary function is its concavity, and this property consequently leads to several important results in finite blocklength analysis. In this paper, we prove that the auxiliary function of a classical-quantum channel also enjoys the same concavity property, extending an earlier partial result to its full generality. We also prove that the auxiliary function satisfies the data-processing inequality, among various other important properties. Furthermore, we show that the concavity property of the auxiliary function enables a geometric interpretation of the random coding exponent and the sphere-packing exponent of a classical-quantum channel. The key component in our proof is an important result from the theory of matrix geometric means.

Exponential decay of matrix Φ-entropies on Markov semigroups with applications to dynamical evolutions of quantum ensembles

In this work, we extend the theory of quantum Markov processes on a single quantum state to a broader theory that covers Markovian evolution of an ensemble of quantum states, which generalizes Lindblad’s formulation of quantum dynamical semigroups. Our results establish the equivalence between an exponential decrease of the matrix Φ-entropies and the Φ-Sobolev inequalities, which allows us to characterize the dynamical evolution of a quantum ensemble to its equilibrium. In particular, we study the convergence rates of two special semigroups, namely, the depolarizing channel and the phase-damping channel. In the former, since there exists a unique equilibrium state, we show that the matrix Φ-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. Howev...

Sphere-packing bound for symmetric classical-quantum channels

“To be considered for the 2017 IEEE Jack Keil Wolf ISIT Student Paper Award.” We provide a sphere-packing lower bound for the optimal error probability in finite blocklengths when coding over a symmetric classical-quantum channel. Our result shows that the pre-factor can be significantly improved from the order of the subexponential to the polynomial, This established pre-factor is arguably optimal because it matches the best known random coding upper bound in the classical case. Our approaches rely on a sharp concentration inequality in strong large deviation theory and crucial properties of the error-exponent function.

Moderate deviations for quantum hypothesis testing and a martingale inequality

“To be considered for the 2017 IEEE Jack Keil Wolf ISIT Student Paper Award.” We study the asymptotic behavior of the type-I error in quantum hypothesis testing when the exponent of the type-II error approaches the quantum relative entropy sufficiently slowly. Our result shows that the moderate deviation principle holds for the testing problem if the quantum relative variance is positive. Our proof strategy employs strong large deviation theory and a martingale inequality.

Moderate deviations for classical-quantum channels

“To be considered for the 2017 IEEE Jack Keil Wolf ISIT Student Paper Award.” We show that the reliable communication through a classical-quantum channel is possible when the transmission rate approaches the channel capacity sufficiently slowly. This scenario exists between the non-vanishing error probability regime, where the rate tends to capacity with a fixed error, and the small error probability regime, where the error vanishes given a rate below capacity. The proof employs a sharp concentration bound in strong large deviation theory, and the asymptotic expansions of the error-exponent functions.