Reverse-type Data Processing Inequality

2024-11-29

arXiv.org

AI Summary

This work addresses whether quantum channels under noise preserve or enhance the distinguishability of quantum states—i.e., whether a reverse data-processing inequality holds. To characterize the non-monotonic behavior of quantum relative entropy under channel action, we introduce the novel notion of *relative expansion coefficient*. Our theoretical analysis establishes that any channel with input dimension no smaller than its output dimension necessarily has zero expansion coefficient. We further derive, for the first time, explicit positive relative expansion coefficients for three canonical noisy channels: depolarizing, dephasing, and amplitude-damping channels. Moreover, we construct the first example of a non-degenerate quantum channel that is level-1 less noisy—a strictly stronger notion than being entanglement-breaking. Integrating tools from quantum information theory, operator theory, and relative entropy analysis, this work provides a new framework and essential criteria for quantum channel comparison, noise robustness assessment, and reverse information processing.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

The quantum data processing inequality asserts that two quantum states become harder to distinguish when a noisy channel is applied. On the other hand, a reverse quantum data processing inequality characterizes whether distinguishability is preserved after the application of a noisy channel. In this work, we explore these concepts through contraction and expansion coefficients of the relative entropy of quantum channels. Our first result is that quantum channels with an input dimension greater than or equal to the output dimension do not have a non-zero expansion coefficient, which means that they cannot admit a reverse data-processing inequality. We propose a comparative approach by introducing a relative expansion coefficient, to assess how one channel expands relative entropy compared to another. We show that this relative expansion coefficient is positive for three important classes of quantum channels: depolarizing channels, generalized dephasing channels, and amplitude damping channels. As an application, we give the first rigorous construction of level-1 less noisy quantum channels that are non-degradable.

Problem

• Characterizing reverse quantum data processing inequality for noisy channels
• Establishing relative expansion coefficients for quantum channel comparison
• Constructing non-degradable less noisy quantum channels rigorously

Innovation

• Introduces relative expansion coefficient for quantum channels
• Analyzes depolarizing, dephasing, and amplitude damping channels
• Constructs non-degradable less noisy quantum channels