For symmetric binary-input channels, every polar-code synthetic channel is equivalent to a mixture of at most polynomially many binary symmetric channels, with an explicit lower bound on the average multiplicity of equal-likelihood outputs.
A partial order for the synthesized channels of a polar code,
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On the Synthetic Channels in Polar Codes over Binary-Input Discrete Memoryless Channels
For symmetric binary-input channels, every polar-code synthetic channel is equivalent to a mixture of at most polynomially many binary symmetric channels, with an explicit lower bound on the average multiplicity of equal-likelihood outputs.