Efficient Approximation of Quantum Channel Capacities

Abstract

We propose an iterative method for approximating the capacity of classical-quantum channels with a discrete input alphabet and a finite dimensional output, possibly 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 $\varepsilon$-close estimate to the capacity, the presented algorithm requires $O(\tfrac{(N \vee M) M^3 \log(N)^{1/2}}{\varepsilon})$, where $N$ denotes the input alphabet size and $M$ the output dimension. We then generalize the method for the task of approximating the capacity of classical-quantum channels with a bounded continuous input alphabet and a finite dimensional output. For channels with a finite dimensional quantum mechanical input and output, the idea of a universal encoder allows us to approximate the Holevo capacity using the same method. In particular, we show that the problem of approximating the Holevo capacity can be reduced to a multidimensional integration problem. For families of quantum channels fulfilling a certain assumption we show that the complexity to derive an $\varepsilon$-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.Comment: 36 pages, 1 figur

A Blahut-Arimoto Type Algorithm for Computing Classical-Quantum Channel Capacity

“…However, it is not a straightforward convex optimization problem. In 2014, Davide Sutter et al [5] promoted an algorithm based on duality of convex programing and smoothing techniques [6] with a complexity of O( (n∨m)m 3 (log n) 1/2 ε ), where n∨m = max{n, m}.…”

“…D(p * ||p t ) < ε) using Algorithm (1) is O(m 3 log ε log (1−δ) ε D(p * ||p N 0 ) ). Usually we do not need ε to be too small (no smaller than 10 −6 ), so in either case, the complexity is better than O( (n∨m)m 3 (log n) 1/2 ε ) in [5] when n ∨ m is big, where n ∨ m = max{n, m}.…”

Efficient Accessible Bounds to the Classical Capacity of Quantum Channels

“…Its analytical value is known mainly for some channels that have the property of additivity, since regularisation is not needed in this case. In fact, in such case the problem is recast to the evaluation of the Holevo capacity [2][3][4], which is a single-letter expression quantifying the maximum information when only product states are sent through the uses of the channel.When a complete knowledge of the channel is available, then several methods can be used to calculate the Holevo capacity [5][6][7][8][9], which is always a lower bound to the ultimate capacity of the channel. In many practical situations, however, a complete knowledge of the kind of noise present along the channel is not available, and sometimes noise can be completely unknown.…”

“…When a complete knowledge of the channel is available, then several methods can be used to calculate the Holevo capacity [5][6][7][8][9], which is always a lower bound to the ultimate capacity of the channel. In many practical situations, however, a complete knowledge of the kind of noise present along the channel is not available, and sometimes noise can be completely unknown.…”

“…where Γ U and Γ V are unitary channels (thus not varying the capacity), and Φ t,Λ is the channel with Λ and t given by Eq. (9). So the Bloch sphere is mapped to an ellipsoid with principal axes parallel to the Cartesian axes, and center shifted by the vector t.…”