Degradable quantum channels are among the only channels whose quantum and private classical capacities are known. As such, determining the structure of these channels is a pressing open question in quantum information theory. We give a comprehensive review of what is currently known about the structure of degradable quantum channels, including a number of new results as well as alternate proofs of some known results. In the case of qubits, we provide a complete characterization of all degradable channels with two dimensional output, give a new proof that a qubit channel with two Kraus operators is either degradable or anti-degradable and present a complete description of anti-degradable unital qubit channels with a new proof. For higher output dimensions we explore the relationship between the output and environment dimensions (dB and dE respectively) of degradable channels. For several broad classes of channels we show that they can be modeled with a environment that is “small ” in the sense dE ≤ dB. Such channels include
We propose the first algorithms with non-asymptotic convergence guarantees for computing the Petz-Augustin capacity, which generalizes the channel capacity and characterizes the optimal error exponent in classical-quantum channel coding. This capacity can be equivalently expressed as the maximization of two generalizations of mutual information: the Petz-Rényi information and the Petz-Augustin information. To maximize the Petz-Rényi information, we show that it corresponds to a convex Hölder-smooth optimization problem, and hence the universal fast gradient method of Nesterov (2015), along with its convergence guarantees, readily applies. Regarding the maximization of the Petz-Augustin information, we adopt a two-layered approach: we show that the objective function is smooth relative to the negative Shannon entropy and can be efficiently optimized by entropic mirror descent; each iteration of entropic mirror descent requires computing the Petz-Augustin information, for which we propose a novel fixed-point algorithm and establish its contractivity with respect to the Thompson metric. Notably, this two-layered approach can be viewed as a generalization of the mirror-descent interpretation of the Blahut-Arimoto algorithm due to He et al. (2024).
We study the computation of the Petz-Augustin mean of order
α∈(0,1)∪(1,∞), defined as the minimizer of a weighted sum of
n Petz-Rényi divergences of order
α over the set of
d-by-
d quantum states, where the Petz-Rényi divergence is a quantum generalization of the classical Rényi divergence. We propose the first algorithm with a non-asymptotic convergence guarantee for solving this optimization problem. The iterates are guaranteed to converge to the Petz-Augustin mean at a linear rate of
O(∣1−1/α∣T) with respect to the Thompson metric for
α∈(1/2,1)∪(1,∞), where
T denotes the number of iterations. The algorithm has an initialization time complexity of
O(nd3) and a per-iteration time complexity of
O(nd2+d3). Two applications follow. The first application concerns the Petz-Augustin capacity of order
α∈(1/2,1), which generalizes the quantum channel capacity and characterizes the optimal error exponent in classical-quantum channel coding. To compute this quantity, we propose the first iterative method with a non-asymptotic convergence guarantee, which uses our algorithm for computing the Petz-Augustin mean as a subroutine. The second application concerns the equilibrium prices in Fisher markets with constant elasticity of substitution (CES) utilities. We establish that when all buyers have CES utilities of common elasticity, the equilibrium prices are equivalent to the Petz-Augustin mean when all quantum states commute. In this case, our proposed algorithm is a tâtonnement process à la Walras. We further extend the algorithm to inhomogeneous Fisher markets, where buyers have CES utilities with different elasticities, and prove a faster convergence rate than existing tâtonnement-type algorithms.