  The fundamental lower bounds of the thermodynamic energy cost (work) needed
for the measurement and the erasure of information are found. The lower bound
for the erasure vindicates the "Landauer's principle" for a special case, but
otherwise implies its breakdown, indicating that no unique relationship exists
between logical reversibility and physical one. Our results constitute the
second law of "information thermodynamics", in which the information content
and thermodynamic variables are treated on an equal footing.
