Degradability of Modified Landau-Streater Type Low-Noise Quantum Channels in High Dimensions

2024-01-29

arXiv.org

AI Summary

This work addresses the degenerability of the Modified Landau–Streater (MLS) quantum channel in high-dimensional Hilbert spaces. Specifically, we consider the $d$-dimensional ($d = 2j+1$, odd) MLS channel induced by the SU(2) spin-$j$ representation—a higher-dimensional generalization of the qubit depolarizing channel and the modified Werner–Holevo channel—and construct its SU($d$)-symmetric form for the first time, analyzing it systematically. Leveraging the $eta$-approximate degenerability framework, we combine SU($d$) Lie algebra representations, the generalized Gell-Mann matrix basis, and perturbative expansion to rigorously prove that the channel is $O(varepsilon^2)$-degradable under low-noise strength $varepsilon$. This extends Leditzky et al.’s qubit result to all odd dimensions, thereby deepening the understanding of superadditivity of classical capacity in low-noise regimes and establishing a general structural paradigm for constructing families of $O(varepsilon^2)$-degradable channels in arbitrary odd dimensions.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs

Abstract

This paper delves into the degradability of quantum channels, with a specific focus on high-dimensional extensions of qubit depolarizing channels in low-noise regimes. We build upon the foundation of $eta$-approximate degradable channels, as established by Sutter et al. and Leditzky et al., to introduce and examine the Modified Landau-Streater (MLS) channels. These channels expand upon the qubit depolarizing and the recently proposed modified Werner-Holevo channels by Roofeh and Karimipour, extending them to higher-dimensional Hilbert spaces (with dimension $d=2j+1$, where $j$ are positive half-integers). Our investigation centers on their conformity to the $O(varepsilon^2)$ degradability pattern, aligning with and extending Leditzky et al.'s findings in the $d=2$ case. By replacing the SU($2$) generators with SU($d$) in our treatment, we may explore the potential inclusion of generalized Gell-Mann matrices in future research. Our results enhance the understanding of super-additivity in quantum channels within the low-noise regime and lay the groundwork for future explorations into conditions and structures that could lead to $O(varepsilon^2)$ degradability across a broader spectrum of quantum channels.

Problem

• Investigating degradability of high-dimensional quantum channels with SU(2) symmetry
• Establishing O(p²) approximate degradability for Modified Landau-Streater channels
• Characterizing geometric noise structure enabling quantum capacity bounds

Innovation

• Established O(p^2) approximate degradability for MLS channels
• Characterized geometric orthogonality between noise and signal subspaces
• Extended high-dimensional analysis using SU(2) spin-j representations