Quantum channels that preserve entanglement
Abstract
Description
Let M and N be full matrix algebras. A unital completely positive (UCP) map ϕ:M\to N is said to preserve entanglement if its inflation ϕ\otimes \id_N : M\otimes N\to N\otimes N has the following property: for every maximally entangled pure state ρof N\otimes N, ρ\circ(ϕ\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.
14 pages, links to references are now fixed
14 pages, links to references are now fixed