Quantum capacity analysis of finite-dimensional lossy channels

TL;DR

This work advances the quantum-capacity analysis of finite-dimensional lossy channels by studying 4-dimensional Multi-level Amplitude Damping (MAD) channels. It develops a comprehensive framework for MADs, including exact composition rules $\Gamma=\Gamma'\Gamma''$, useful decompositions into level-by-level decays and single-decay blocks, and an explicit expression for the complementary MAD channel. A central achievement is the complete characterization of antidegradability in the MAD parameter space, together with a practical recipe to determine degradability regions. Building on this, the authors present a procedure to compute quantum capacity in degradable regions, extend it to non-degradable cases via unitary covariances and monotonicity, and illustrate the method with a detailed 4D example. They also propose a conjecture about per-level exclusion under partial antidegradability and validate it in MAD3, with implications for the private quantum capacity and encoding strategies across MAD families.

Abstract

Traditionally, Quantum Information, and Quantum Communication specifically, have been focused on qubit-based architectures. Recent results, however, highlighted that higher dimensional architectures (qudit-based) may present advantages both in terms of communication and computation; a family of channels called Multi-level Amplitude Damping (MAD) channels, which are a possible qudit generalization of the well known Amplitude Damping Channels, is able to model energy decay processes that may happen during signal transmission. In this work, the Quantum Capacity of 4-dimensional MAD's is studied, relying on a technique for computing it even outside of degradable and antidegradable conditions. We also characterized the complete region of antidegradability and degradability in the parameter space for a generic d-dimensional MAD using both analytical and semi-numerical methods.

Figures (16)

• Figure 1.1: MAD channels represent decay processes, where each level of a system has a fixed probability of decaying onto a lower level. Here, a schematic depiction of a $d$-dimensional MAD is reported.
• Figure 2.1: Visual representation of the decomposition of a $4$-dimensional MAD channel using \ref{['equation:MADdecomposition1']}, to be read from left to right in "chronological" order. Thanks to \ref{['equation:MADdecomposition1']}, one could interpret a generic $4$-dimensional MAD channel as the action of a single decay from $\ket{1}$ to $\ket{0}$, followed by a double decay from $\ket{2}$ to $\ket{1}$ and $\ket{0}$, followed by a triple decay from $\ket{3}$ to $\ket{2}$, $\ket{1}$ and $\ket{0}$.
• Figure 5.1: Depiction of the "problematic" ladder-like structure that renders MAD's non degradable, in the case of $d=4$.
• Figure 6.1: Graphic depiction and parameter space of the MAD with transition matrix defined in \ref{['equation:def:examplegamma']}
• Figure 6.2: First step of extension of the computation of the quantum capacity for the MAD with transition matrix \ref{['equation:def:examplegamma']}.