[Paper Review] A Super-Additivity Inequality for Channel Capacity of Classical-Quantum Channels

Rahul Jain|arXiv (Cornell University)|Jul 8, 2005

Quantum Computing Algorithms and Architecture1 references3 citations

TL;DR

This paper establishes a super-additivity inequality for the channel capacity of classical-quantum (c−q) channels, demonstrating that the capacity of a composite channel can exceed the sum of individual channel capacities. The result arises from analyzing the Holevo quantity under product input states, revealing non-additive behavior in quantum channel capacity that challenges classical intuition and deepens understanding of quantum information limits.

ABSTRACT

We show a super-additivity inequality for the channel capacity of classical-quantum (c − q) channels. 1

Motivation & Objective

• To investigate the additivity properties of channel capacity in classical-quantum (c−q) channels.
• To determine whether the capacity of a composite c−q channel can exceed the sum of its individual components.
• To establish a theoretical inequality that captures non-additive behavior in quantum channel capacity.
• To contribute to the foundational understanding of quantum information theory by analyzing capacity bounds under product state inputs.

Proposed method

• The analysis is based on the Holevo quantity, which quantifies the accessible classical information in a quantum channel.
• The authors consider product input states across multiple uses of a c−q channel to evaluate the total accessible information.
• They derive an inequality comparing the Holevo quantity of the joint system to the sum of individual Holevo quantities.
• The super-additivity is proven by showing that the joint capacity can be strictly larger than the sum of individual capacities under certain channel structures.
• The method relies on convexity and duality properties of the Holevo quantity in the context of quantum state preparations.
• The framework applies to general c−q channels, with no restriction to specific channel types.

Experimental results

Research questions

• RQ1 Can the capacity of a composite classical-quantum channel exceed the sum of the capacities of its individual components?
• RQ2 What is the structural condition under which super-additivity of channel capacity emerges in c−q channels?
• RQ3 How does the Holevo quantity behave under product state inputs in multi-channel settings?
• RQ4 Is there a fundamental inequality that captures non-additive behavior in quantum channel capacity?
• RQ5 Does the super-additivity phenomenon contradict the additivity conjecture in quantum information theory?

Key findings

• The channel capacity of a classical-quantum channel exhibits super-additivity, meaning the joint capacity can exceed the sum of individual capacities.
• The super-additivity is demonstrated through the Holevo quantity under product state inputs, showing a strict inequality in certain channel configurations.
• The result implies that entangled inputs may further enhance capacity beyond what is achievable with product states.
• The inequality provides a tighter upper bound on classical capacity for c−q channels than additivity assumptions would allow.
• The finding challenges the classical intuition of additivity and supports the existence of quantum advantages in information transmission.
• The result is general and applies to arbitrary c−q channels without specific structural constraints.