  Let M and N be full matrix algebras. A unital completely positive (UCP) map
\phi:M\to N is said to preserve entanglement if its inflation \phi\otimes \id_N
: M\otimes N\to N\otimes N has the following property: for every maximally
entangled pure state \rho of N\otimes N, \rho\circ(\phi\otimes \id_N) is an
entangled state of M\otimes N.
  We show that there is a dichotomy in that every UCP map that is not
entanglement breaking in the sense of Horodecki-Shor-Ruskai must preserve
entanglement, and that entanglement preserving maps of every possible rank
exist in abundance. We also show that with probability 1, {\em all} UCP maps of
relatively small rank preserve entanglement, but that this is not so for UCP
maps of maximum rank.
