Algorithmic complexity of quantum capacity

TL;DR

This paper investigates whether the theoretical quantum capacity aligns with practical computability by introducing an algorithmic approach that approximates quantum states and channels with rational numbers, ultimately showing their equivalence.

Contribution

It develops an algorithmic framework for quantum capacity, demonstrating that the standard quantum capacity can be characterized through semi-computability and algorithmic entropies.

Findings

Algorithmic quantum capacity equals the standard quantum capacity.

Introduces semi-computability to approximate quantum states and channels.

Defines and analyzes algorithmic quantum entropies.

Abstract

Recently the theory of communication developed by Shannon has been extended to the quantum realm by exploiting the rules of quantum theory. This latter stems on complex vector spaces. However complex (as well as real) numbers are just idealizations and they are not available in practice where we can only deal with rational numbers. This fact naturally leads to the question of whether the developed notions of capacities for quantum channels truly catch their ability to transmit information. Here we answer this question for the quantum capacity. To this end we resort to the notion of semi-computability in order to approximately (by rational numbers) describe quantum states and quantum channel maps. Then we introduce algorithmic entropies (like algorithmic quantum coherent information) and derive relevant properties for them. Finally we define algorithmic quantum capacity and prove that it equals the standard one.