Entanglement measures and the Hilbert-Schmidt distance
  In order to construct a measure of entanglement on the basis of a ``distance'' between two states, it is one of desirable properties that the ``distance'' is nonincreasing under every completely positive trace preserving map. Contrary to a recent claim, this letter shows that the Hilbert-Schmidt distance does not have this property. 
Masanao Ozawa (Nagoya University)