Geometric Rényi Divergence and its Applications in Quantum Channel Capacities

Abstract

We present a systematic study of the geometric Rényi divergence (GRD), also known as the maximal Rényi 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 Rényi 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 Rényi 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.

References in corpus (20)

• Fault-tolerant quantum computation with high threshold in two dimensions
• The Quantum Chernoff Bound
• A complete family of separability criteria
• Magic state distillation with low overhead
• A Sharp Fannes-type Inequality for the von Neumann Entropy
• The Chernoff lower bound for symmetric quantum hypothesis testing
• Quantifying the magic of quantum channels
• Quantifying magic for multi-qubit operations
• Fault-Tolerant Computing With Biased-Noise Superconducting Qubits
• Unbounded number of channel uses are required to see quantum capacity
• Efficiently computable bounds for magic state distillation
• The quantum capacity with symmetric side channels
• A chain rule for the quantum relative entropy
• The sum-of-squares hierarchy on the sphere, and applications in quantum information theory
• Entanglement of a bipartite channel
• Fault-Tolerant Postselected Quantum Computation: Threshold Analysis
• Fundamental limits on the capacities of bipartite quantum interactions
• Additive Extensions of a Quantum Channel
• A strengthened data processing inequality for the Belavkin-Staszewski relative entropy
• Information-theoretic aspects of the generalized amplitude damping channel

Cited by in corpus (5)

• A chain rule for the quantum relative entropy
• No-go theorems for quantum resource purification II: new approach and channel theory
• Exponential decay of mutual information for Gibbs states of local Hamiltonians
• Defining quantum divergences via convex optimization
• Black hole microstates vs. the additivity conjectures