Minimal energy cost for thermodynamic information processing: measurement and information erasure.

The fundamental lower bounds on the thermodynamic energy cost of measurement and information erasure are determined. The lower bound on the erasure validates Landauer's principle for a symmetric memory; for other cases, the bound indicates the breakdown of the principle. Our results constitute the second law of "information thermodynamics," in which information content and thermodynamic variables are treated on an equal footing.