How to prove the Maxwell conjecture via spatial coupling — A proof of concept

Investigations on spatially coupled codes have lead to the conjecture that, in the infinite size limit, the average input-output conditional entropy for spatially coupled low-density parity-check ensembles, over binary memoryless symmetric channels, equals the entropy of the underlying individual ensemble. We give a self-contained proof of this conjecture for the case when the variable degrees have a Poisson distribution and all check degrees are even. The ingredients of the proof are the interpolation method and the Nishimori identities. We explain why this result is an important step towards proving the Maxwell conjecture in the theory of low-density parity-check codes.