  Reversing the effects of a quantum evolution, for example as is done in error
correction, is an important task for controlling quantum systems in order to
produce reliable quantum devices. When the evolution is governed by a
completely positive map, there exist reversibility conditions, known as the
quantum error correcting code conditions, which are necessary and sufficient
conditions for the reversibility of a quantum operation on a subspace, the code
space. However, if we suppose that the evolution is not described by a
completely positive map, necessary and sufficient conditions are not known.
Here we consider evolutions that do not necessarily correspond to a completely
positive map. We prove that the completely positive map error correcting code
conditions can lead to a code space that is not in the domain of the map,
meaning that the output of the map is not positive. A corollary to our theorem
provides a class of relevant examples. Finally, we provide a set of sufficient
conditions that will enable the use of quantum error correcting code conditions
while ensuring positivity.
