Efficient Approximation of Quantum Channel Capacities

David Sutter, Tobias Sutter, Peyman Mohajerin Esfahani, Renato Renner

Abstract: We propose an iterative method for approximating the capacity of classical-quantum channels with a discrete input alphabet and a finite-dimensional output under additional constraints on the input distribution. Based on duality of convex programming, we derive explicit upper and lower bounds for the capacity. To provide an additive ε-close estimate to the capacity, the presented algorithm requires O((N ν M)M <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> log(N) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup> ε <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> ) steps, where N denotes the input alphabet size and M denotes the output dimension. We then generalize the method to the task of approximating the capacity of classical-quantum channels with a bounded continuous input alphabet and a finite-dimensional output. This, using the idea of a universal encoder, allows us to approximate the Holevo capacity for channels with a finite-dimensional quantum mechanical input and output. In particular, we show that the problem of approximating the Holevo capacity can be reduced to a multi-dimensional integration problem. For certain families of quantum channels, we prove that the complexity to derive an additive ε-close solution to the Holevo capacity is subexponential or even polynomial in the problem size. We provide several examples to illustrate the performance of the approximation scheme in practice.