Structural Conditions for Quadratic Degradability in Symmetric Quantum Channels

TL;DR

The paper addresses the quantum capacity of high-dimensional, symmetry-rich channels by analyzing SU(2)-covariant MLS channels. It develops an explicit degrading map tied to the Casimir operator and proves a universal η = O(p^2) bound for approximate degradability, valid for all spins j, supported by a geometric orthogonality of noise and identity. The work identifies two algebraic features—noise tracelessness and normal eigen-invariance—as the mechanism behind quadratic degradation and extends the framework to generalized Pauli channels, yielding tighter low-noise capacity estimates and a general structural condition for quadratic degradability. Numerical analysis corroborates the quadratic scaling and demonstrates practical capacity bounds via the coherent information, highlighting the role of symmetry as a resource in quantum communication theory.

Abstract

This paper studies the role of symmetry in determining the low-noise behavior of high-dimensional quantum channels, with a focus on the modified Landau--Streater (MLS) family derived from spin-$j$ representations of $SU(2)$. We show that these channels exhibit a quadratic scaling of the approximate degradability parameter, $η= O(p^2)$, for every dimension $d=2j+1$, using an explicit degrading map whose structure is dictated by the $SU(2)$ Casimir operator. We further identify two algebraic features underlying this behavior: the tracelessness of the noise operators, which enforces a geometric orthogonality between signal and noise components, and an eigen-invariance property that preserves this structure under the channel action. These observations lead to a general sufficient condition under which a channel displays quadratic suppression of non-degradability, placing the MLS and generalized Pauli channels within a common analytical framework. Overall, the results clarify why certain symmetric noise models admit tighter low-noise capacity estimates and suggest a structural perspective for analyzing approximate degradability in higher dimensions.

Figures (2)

• Figure 1: Comparison of the single-letter coherent information $I_c(\pi)$ (solid blue) and the approximate degradability lower bounds for the spin-1 MLS channel ($d=3$). The red dashed line represents the lower bound derived using Theorem 1 of leditzky2018prl with $\eta \sim O(p^2)$, which tracks the coherent information closely. The generic scaling $\eta \sim O(p^{1.5})$ (black dotted line), typical for non-symmetric channels, yields a significantly looser bound. Note that since $P(\mathcal{N}) \ge Q(\mathcal{N})$, this lower bound applies to both quantum and private capacities.
• Figure 2: Log-log plot of the approximate degradability parameter $\eta$ versus the noise parameter $p$. The data points (squares for $j=1$, triangles for $j=3/2$) perfectly align with a slope of 2, numerically validating the $\eta \sim O(p^2)$ theorem. The dashed line represents the generic scaling for non-symmetric channels ($O(p^{1.5})$), highlighting the significant advantage achieved by the symmetry-protected MLS channel.

Theorems & Definitions (10)

• Lemma 1: From leditzky2018prl
• Definition 1: Approximate Degradability
• Theorem 1: Geometric Orthogonality
• proof
• Corollary 1: $\varepsilon$-perfectness of MLS Channels
• Theorem 2: $O(p^2)$ Degradability
• Proposition 1: Sufficient Condition for Quadratic Degradability
• proof
• Lemma 2
• proof