  In open quantum systems, it is known that if the system and environment are
in a product state, the evolution of the system is given by a linear completely
positive (CP) Hermitian map. CP maps are a subset of general linear Hermitian
maps, which also include non completely positive (NCP) maps. NCP maps can arise
in evolutions such as non-Markovian evolution, where the CP divisibility of the
map (writing the overall evolution as a composition of CP maps) usually fails.
Positive but NCP maps are also useful as entanglement witnesses. In this paper,
we focus on transforming an initial NCP map to a CP map through composition
with the asymmetric depolarizing map. We use separate asymmetric depolarizing
maps acting on the individual subsystems.
  Previous work have looked at structural physical approximation (SPA), which
is a CP approximation of a NCP map using a mixture of the NCP map with a
completely depolarizing map. We prove that the composition can always be made
CP without completely depolarizing in any direction. It is possible to
depolarize less in some directions. We give the general proof by using the Choi
matrix and an isomorphism from a maximally entangled two qudit state to a set
of qubits. We also give measures that describe the amount of disturbance the
depolarization introduces to the original map. Given our measures, we show that
asymmetric depolarization has many advantages over SPA in preserving the
structure of the original NCP map. Finally, we give some examples. For some
measures and examples, completely depolarizing (while not necessary) in some
directions can give a better approximation than keeping the depolarizing
parameters bounded by the required depolarization if symmetric depolarization
is used.
