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. (C) 2000 Elsevier Science B.V. All rights reserved.