Quantum Channels That Preserve Entanglement

Let M and N be full matrix algebras. A unital completely positive (UCP) map φ : M → N is said to preserve entanglement if its inflation φ ⊗ idN : M ⊗ N → N ⊗ N has the following property: for every maximally entangled pure state ρ of N ⊗ N , ρ ◦ (φ ⊗ idN) is an entangled state of M ⊗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, all UCP maps of relatively small rank preserve entanglement, but that this is not so for UCP maps of maximum rank.