Submartingale Property of E_0 Under The Polarization Transformations

TL;DR

This paper proves a submartingale property of a channel parameter under polarization transformations, showing that the sum of the parameter for the transformed channels exceeds twice its original value.

Contribution

It establishes a new inequality demonstrating the submartingale nature of E_0 under channel polarization transformations for binary input channels.

Findings

E_0(W^-) + E_0(W^+) ≥ 2 E_0(W) for all ρ ≥ 0

The relation holds for any binary input discrete memoryless channel W

Supports the theoretical understanding of polarization process

Abstract

We prove that the relation $E_0(\rho, W^{-}) + E_0(\rho, W^{+}) \geq 2 E_0(\rho, W)$ holds for any binary input discrete memoryless channel $W$, and $\rho \geq 0$.