arXiv:quant-ph/0606176v1 21 Jun 2006 On quantum positive dynamical maps. W. A. Majewski 1 Institute of Theoretical Physics and Astrophysics Gda´ nsk University Wita Stwosza 57 80-952 Gda´ nsk, Poland E-mail address: fizwam@univ.gda.pl Abstract: The new arguments indicating that non-completely positive maps can describe open quantum evolution are given. 1 This work was partially supported by KBN grant PB/1490/PO3/2003/25 and SCALA (IST-2004-015714) 1