Computing Classical-Quantum Channel Capacity Using Blahut–Arimoto Type Algorithm: A Theoretical and Numerical Analysis

Based on Arimoto’s work in 1972, we propose an iterative algorithm for computing the capacity of a discrete memoryless classical-quantum channel with a finite input alphabet and a finite dimensional output, which we call the Blahut–Arimoto algorithm for classical-quantum channel, and an input cost constraint is considered. We show that, to reach ε accuracy, the iteration complexity of the algorithm is upper bounded by log n log ε ε where n is the size of the input alphabet. In particular, when the output state { ρ x } x ∈ X is linearly independent in complex matrix space, the algorithm has a geometric convergence. We also show that the algorithm reaches an ε accurate solution with a complexity of O ( m 3 log n log ε ε ) , and O ( m 3 log ε log ( 1 − δ ) ε D ( p * | | p N 0 ) ) in the special case, where m is the output dimension, D ( p * | | p N 0 ) is the relative entropy of two distributions, and δ is a positive number. Numerical experiments were performed and an approximate solution for the binary two-dimensional case was analysed.

A Formal Model for Semantic Computing Based on Generalized Probabilistic Automata

In most previous research, “semantic computing” refers to computational implementations of semantic reasoning. It lacks support from the formal theory of computation. To provide solid foundations for semantic computing, researchers propose a different understanding of semantic computing based on finite automata. This approach provides a computer theoretical approach to semantic computing. But finite automata are not capable enough to deal with imprecise knowledge. Therefore, in this paper, we provide foundations for semantic computing based on probabilistic automata. Even though traditional probabilistic automata can handle imprecise knowledge, their limitation resides in their being defined on a fixed finite input alphabet. This deeply restricts the abilities of automata. In this paper, we rebuild traditional probabilistic automata for semantic computing. Furthermore, our new probabilistic automata are robust enough to handle any alphabet as input. They have better performances in many applications. We provide an application for weather forecasting, a domain for which traditional probabilistic automata are not effective due to their finite input alphabet. Our new probabilistic automata can overcome these limitations.

A NOTE ON LIMITED PUSHDOWN ALPHABETS IN STATELESS DETERMINISTIC PUSHDOWN AUTOMATA

Recently, an infinite hierarchy of languages accepted by stateless deterministic pushdown automata has been established based on the number of pushdown symbols. However, the witness language for the nth level of the hierarchy is over an input alphabet with 2(n − 1) elements. In this paper, we improve this result by showing that a binary alphabet is sufficient to establish this hierarchy. As a consequence of our construction, we solve the open problem formulated by Meduna et al. Then we extend these results to m-state realtime deterministic pushdown automata, for all m ≥ 1. The existence of such a hierarchy for m-state deterministic pushdown automata is left open.