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January 17, 20260 citationsOpen Access

Jensen–Shannon Contraction for Symmetric Channels: The Binary Case is Unique

Key Points

This work aims to establish the exact contraction coefficients for the Jensen–Shannon divergence in symmetric channels, focusing on the binary case.
Analytic proof of contraction coefficients for Jensen–Shannon divergence.
Examination of binary symmetric channel characteristics.
Comparison of Jensen–Shannon and chi-squared contraction coefficients across symmetric channels.
The contraction coefficient for the binary symmetric channel is exactly (1-2δ)^2.
A strict separation between Jensen–Shannon and chi-squared coefficients is established for channels with m≥3.
Identified the binary symmetric channel as the unique case where both coefficients coincide across all noise levels.

Abstract

This work establishes exact strong data-processing (contraction) coefficients for the Jensen–Shannon divergence across symmetric discrete channels. A complete analytic proof shows that, for the binary symmetric channel, the JSD contraction coefficient is exactly \ ( (1-2) ²\), even though the generating function of JSD is not operator convex and standard equivalence theorems are therefore inapplicable. For all \ (m 3\) symmetric channels, we rigorously demonstrate a strict separation between the Jensen–Shannon and \ (²\) contraction coefficients. This identifies the binary symmetric channel as the unique nontrivial symmetric channel family where these coefficients coincide for all noise levels.

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