[Paper Review] Information Geometric Superactivation of Asymptotic Quantum Capacity and Classical Zero-Error Capacity of Zero-Capacity Quantum Channels

László Gyöngyösi|arXiv (Cornell University)|Jun 25, 2012

Quantum Computing Algorithms and Architecture13 references3 citations

TL;DR

This paper presents an information geometric framework that explains and algorithmically identifies superactivation in zero-capacity quantum channels—where two channels with individually zero asymptotic quantum or classical zero-error capacity jointly enable non-zero communication. The key contribution is proving that superactivation arises from intrinsic geometric structures in quantum state space, offering a systematic method to detect and classify such effects beyond ad hoc examples.

ABSTRACT

The superactivation of zero-capacity quantum channels makes it possible to use two zero-capacity quantum channels with a positive joint capacity at the output. Currently, we have no theoretical background for describing all possible combinations of superactive zero-capacity channels; hence, there may be many other possible combinations. In this PhD Thesis I provide an algorithmic solution to the problem of superactivation and prove that superactivation effect is rooted in information geometric issues.

Motivation & Objective

• To develop a theoretical and algorithmic framework for identifying all possible superactivation effects in zero-capacity quantum channels.
• To address the lack of a general theory explaining which combinations of zero-capacity channels can jointly achieve positive capacity.
• To establish that superactivation is not a sporadic phenomenon but rooted in deep information-geometric properties of quantum channels.
• To provide a comprehensive classification of superactivation via geometric invariants in quantum state manifolds.

Proposed method

• The author employs information geometry to model the space of quantum channels as Riemannian manifolds, using the quantum Fisher information metric to analyze curvature and structure.
• A novel algorithm is developed to scan the parameter space of channel pairs to detect superactivation by identifying non-vanishing joint capacities.
• The method leverages the duality between quantum capacity and zero-error capacity, using entanglement-assisted and classical zero-error capacity criteria in a unified geometric framework.
• Geometric invariants such as sectional curvature and holonomy are computed to detect non-trivial channel interactions that signal superactivation.
• The approach uses the asymptotic quantum capacity and classical zero-error capacity as primary measures, with capacity thresholds defined via relative entropy and fidelity-based bounds.
• The framework integrates results from quantum information theory and differential geometry to derive necessary and sufficient conditions for superactivation in terms of manifold topology and metric structure.

Experimental results

Research questions

• RQ1 Which combinations of zero-capacity quantum channels can exhibit superactivation, and what structural conditions enable this phenomenon?
• RQ2 Can superactivation be systematically predicted rather than discovered case by case?
• RQ3 What is the underlying geometric origin of superactivation in quantum channels, and how does it relate to quantum state space curvature?
• RQ4 How do the asymptotic quantum capacity and classical zero-error capacity interrelate in the context of information-geometric superactivation?
• RQ5 Can a unified geometric invariant be derived to classify all superactivatable channel pairs?

Key findings

• Superactivation in zero-capacity quantum channels is not an isolated quantum phenomenon but is fundamentally rooted in the information-geometric structure of the underlying quantum state space.
• The proposed algorithm successfully identifies superactivation effects across a broad class of channel pairs, demonstrating that such effects are more widespread than previously known.
• Geometric invariants such as non-zero sectional curvature in the quantum channel manifold serve as reliable indicators of superactivation potential.
• The study establishes a direct link between the non-trivial topology of the quantum state space and the emergence of joint capacity where individual channels fail.
• The framework reveals that superactivation is detectable via metric properties like quantum Fisher information, enabling a predictive and scalable method for identifying new superactivatable channel pairs.
• The results confirm that both asymptotic quantum capacity and classical zero-error capacity can be superactivated simultaneously in certain channel combinations, extending the scope of the phenomenon.