We study channel simulation under common randomness assistance in the finite-blocklength regime and identify the smooth channel max-information as a linear program one-shot converse on the minimal simulation cost for fixed error tolerance. We show that this one-shot converse can be achieved exactly using no-signaling-assisted codes, and approximately achieved using common randomness-assisted codes. Our one-shot converse thus takes on an analogous role to the celebrated meta-converse in the complementary problem of channel coding, and we find tight relations between these two bounds. We asymptotically expand our bounds on the simulation cost for discrete memoryless channels, leading to the second-order as well as the moderate-deviation rate expansion, which can be expressed in terms of the channel capacity and channel dispersion known from noisy channel coding. Our bounds imply the well-known fact that the optimal asymptotic rate of one channel to simulate another under common randomness assistance is given by the ratio of their respective capacities. Additionally, our higher-order asymptotic expansion shows that this reversibility falls apart in the second order. Our techniques extend to discrete memoryless broadcast channels. In stark contrast to the elusive broadcast channel capacity problem, we show that the reverse problem of broadcast channel simulation under common randomness assistance allows for an efficiently computable single-letter characterization of the asymptotic rate region in terms of the broadcast channel’s multipartite mutual information.

We study the problem of random-assisted simulation of discrete broadcast channel in one-shot and i.i.d. setups. We derive one-shot inner and outer bounds of the set of attainable message-size pairs for simulating WYZ|Xwithin some total variation distance (TVD) tolerance of ϵ. The inner bounds are based on the bipartite convex split lemma. Whereas the outer bounds are based on the properties of the multi-partite max information. Using these bounds, we establish a single-letter expression of the simulation region of a broadcast channel.

The moderate deviation regime is concerned with the finite block length trade-off between communication cost and error for information processing tasks in the asymptotic regime, where the communication cost approaches a capacity-like quantity and the error vanishes at the same time. We find exact characterisations of these trade-offs for a variety of fully quantum communication tasks, including quantum source coding, quantum state splitting, entanglement-assisted quantum channel coding, and entanglement-assisted quantum channel simulation. The main technical tool we derive is a tight relation between the partially smoothed max-information and the hypothesis testing relative entropy. This allows us to obtain the expansion of the partially smoothed max-information for i.i.d. states in the moderate deviation regime.

We study the problem of one-shot channel simulation of DMCs with unlimited shared randomness. For any fixed tolerance measured in total variational distance, we propose an achievability bound and a converse bound on the size of the code to simulate the channel. The achievability bound utilizes the convex split lemma, whereas the converse bound is the result of the relationships between smoothed max-divergences and the max-mutual information. The achievability proof does not rely on a "universal state" (compared with some previous related works), and provides a tighter bound. Using the two bounds, we also provide an alternative proof to the reverse Shannon theorem.

The capacity of noisy quantum channels characterizes the highest rate at which information can be reliably transmitted and it is therefore of practical as well as fundamental importance. Capacities of classical channels are computed using alternating optimization schemes, called Blahut-Arimoto algorithms. In this work, we generalize classical Blahut-Arimoto algorithms to the quantum setting. In particular, we give efficient iterative schemes to compute the capacity of channels with classical input and quantum output, the quantum capacity of less noisy channels, the thermodynamic capacity of quantum channels, as well as the entanglement-assisted capacity of quantum channels. We give rigorousa priorianda posterioribounds on the estimation error by employing quantum entropy inequalities and demonstrate fast convergence of our algorithms in numerical experiments.

We generalize alternating optimization algorithms of Blahut-Arimoto type to the quantum setting. In particular, we give iterative algorithms to compute the mutual information of quantum channels, the thermodynamic capacity of quantum channels, the coherent information of less noisy quantum channels, and the Holevo quantity of classical-quantum channels. Our convergence analysis is based on quantum entropy inequalities and leads to a priori additive ε-approximations after O (ε-1log N) iterations, where N denotes the input dimension of the channel. We complement our analysis with an a posteriori stopping criterion which allows us to terminate the algorithm after significantly fewer iterations compared to the a priori criterion in numerical examples. Finally, we discuss heuristics to accelerate the convergence.