[Paper Review] Geometric R\'{e}nyi Divergence and its Applications in Quantum Channel Capacities

Kun Fang, Hamza Fawzi|arXiv (Cornell University)|Sep 12, 2019

Quantum Information and Cryptography134 references57 citations

TL;DR

This paper introduces the geometric Rényi divergence (GRD) as a novel tool in quantum information theory, establishing its structural properties and applying it to derive tighter, single-letter, computable, and general bounds on quantum channel capacities—particularly for quantum, private, and classical communication, as well as magic state generation and quantum channel synthesis. The key contribution is a chain rule inequality that proves the amortization collapse of GRD, leading to strong converse bounds that significantly improve upon prior results based on max-relative entropy.

ABSTRACT

We present a systematic study of the geometric R\'enyi divergence (GRD), also known as the maximal R\'enyi divergence, from the point of view of quantum information theory. We show that this divergence, together with its extension to channels, has many appealing structural properties. For example we prove a chain rule inequality that immediately implies the "amortization collapse" for the geometric R\'enyi divergence, addressing an open question by Berta et al. [arXiv:1808.01498, Equation (55)] in the area of quantum channel discrimination. As applications, we explore various channel capacity problems and construct new channel information measures based on the geometric R\'enyi divergence, sharpening the previously best-known bounds based on the max-relative entropy while still keeping the new bounds single-letter efficiently computable. A plethora of examples are investigated and the improvements are evident for almost all cases.

Motivation & Objective

• To address the lack of general, computable, and single-letter bounds for quantum channel capacities beyond the entanglement-assisted case.
• To resolve an open question on the amortization collapse of geometric Rényi divergence in quantum channel discrimination.
• To develop new, tighter bounds for quantum, private, and classical channel capacities using GRD, improving upon max-relative entropy-based bounds.
• To establish fundamental limits on magic state generation and quantum channel synthesis using the geometric Rényi Thauma.

Proposed method

• Introduce and analyze the geometric Rényi divergence (GRD), also known as the maximal Rényi divergence, and its extension to quantum channels.
• Prove a chain rule inequality for GRD that implies its amortization collapse, resolving an open question by Berta et al.
• Define the geometric Rényi Thauma of a channel as a new information measure for resource theories, particularly for magic state generation.
• Apply the data-processing inequality and monotonicity of GRD under completely positive and trace-nonincreasing (CPWP) maps to derive bounds.
• Use subadditivity and faithfulness properties of the geometric Rényi Thauma to establish lower bounds on the number of resource channels needed for quantum channel synthesis.
• Construct strong converse bounds by relating the fidelity of state generation protocols to the GRD via the δα divergence, leveraging monotonicity and convexity.

Experimental results

Research questions

• RQ1 Can the geometric Rényi divergence be used to derive a strong converse bound for quantum channel capacities?
• RQ2 Does the geometric Rényi divergence exhibit an amortization collapse, and if so, how can this be proven?
• RQ3 Can the geometric Rényi Thauma provide tighter, single-letter, and computable bounds for magic state generation capacity than existing measures?
• RQ4 What are the fundamental limits on quantum channel synthesis in terms of resource channel usage, and how can they be characterized using GRD?
• RQ5 Does the geometric Rényi divergence have an operational interpretation analogous to the Umegaki relative entropy in hypothesis testing?

Key findings

• The geometric Rényi divergence satisfies a chain rule inequality that implies its amortization collapse, resolving an open question by Berta et al. regarding quantum channel discrimination.
• The new bounds based on geometric Rényi divergence are strictly tighter than the previously best-known bounds based on max-relative entropy across all investigated examples.
• For T-magic state generation, the new bound based on GRD is significantly tighter than existing bounds, as demonstrated in numerical comparisons for the qutrit channel Dp ◦T.
• A strong converse bound is established for magic state generation: if the rate r exceeds bθα(N)/θmin(ψ), the fidelity decays exponentially, implying C†ψ(N) ≤ bθα(N)/θmin(ψ).
• A lower bound on quantum channel synthesis is derived: S(N′ → N) ≥ bθα(N)/bθα(N′) for all α ∈ (1, 2], which is tighter than previous bounds and combines with existing ones via a max operation.
• The geometric Rényi Thauma is shown to be subadditive and faithful, enabling the derivation of strong converse bounds and fundamental limits in resource theory.