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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., Comment: 13 pages, 1 figure, comments welcome! v2: Introduction enhanced

This paper introduces a novel mathematical model for Molecular Communication (MC) systems, utilizing First Arrival Position (FAP) as a fundamental mode of information transmission. We address two critical challenges: the characterization of FAP density and the establishment of capacity bounds for channels with vertically-drifted FAP. Our method relate macroscopic Partial Differential Equation (PDE) models to microscopic Stochastic Differential Equation (SDE) models, resulting in a precise expression that links FAP density with elliptic-type Green's function. This formula is distinguished by its wide applicability across any spatial dimensions, any drift directions, and various receiver geometries. We demonstrate the practicality of our model through case studies: 2D and 3D planar receivers. The accuracy of our formula is also validated by particle-based simulations. Advancing further, the explicit FAP density forms enable us to establish closed-form upper and lower bounds for the capacity of vertically-drifted FAP channels under a second-moment constraint, significantly advancing the understanding of FAP channels in MC systems., Comment: 16 pages, 9 figures, 1 table; this manuscript (v3) is submitted to IEEE Transactions on Communications and currently is under minor revision

In this paper, we delve into the capacity problem of additive vertically-drifted first arrival position noise channel, which models a communication system where the position of molecules is harnessed to convey information. Drawing inspiration from the principles governing vector Gaussian interference channels, we examine this capacity problem within the context of a covariance constraint on input distributions. We offer analytical upper and lower bounds on this capacity for a three-dimensional spatial setting. This is achieved through a meticulous analysis of the characteristic function coupled with an investigation into the stability properties. The results of this study contribute to the ongoing effort to understand the fundamental limits of molecular communication systems., Comment: 6 pages, 1 figure

Recent advancements in understanding the impulse response of the first arrival position (FAP) channel in molecular communication (MC) have illuminated its Shannon capacity. While Lee et al. shed light on FAP channel capacities with vertical drifts, the zero-drift scenario remains a conundrum, primarily due to the challenges associated with the heavy-tailed Cauchy distributions whose first and second moments do not exist, rendering traditional mutual information constraints ineffective. This paper unveils a novel characterization of the zero drift FAP channel capacity for both 2D and 3D. Interestingly, our results reveal a 3D FAP channel capacity that is double its 2D counterpart, hinting at a capacity increase with spatial dimension growth. Furthermore, our approach, which incorporates a modified logarithmic constraint and an output signal constraint, offers a simplified and more intuitive formula (similar to the well-known Gaussian case) for estimating FAP channel capacity., Comment: 8 pages, 2 figures, 1 table

This paper introduces a novel unified framework for calculating the first arrival position (FAP) density in diffusion-based molecular communication (MC) systems with fully-absorbing receivers, applicable to any spatial dimension or receiver shape. We validate the effectiveness of our proposed method by presenting two concrete examples from the MC literature: the 2D line receiver and 3D spherical receiver. Our approach successfully reproduces existing results on FAP density and further enhances the original formula for the 2D line receiver by accommodating arbitrary drift directions. The proposed framework offers a comprehensive and versatile solution for determining FAP density in diffusive MC channels with fully-absorbing receivers, paving the way for more robust and efficient system designs., Comment: 6 pages, 2 figures, conference