Gluing of completely positive maps

Gluings of completely positive maps (CPMs) are defined and investigated. As a brief description of this concept consider a pair of `evolution machines', each with the ability to evolve the internal state of a `particle' inserted into its input. Each of these machines is characterized by a channel describing the operation the internal state has experienced when the particle is returned at the output. Suppose a particle is put in a superposition between the input of the first and the second machine. Here it is shown that the total evolution caused by a pair of such devices is not uniquely determined by the channels of the two machines. Such `global' channels describing the machine pair are examples of gluings of the two single machine channels. Under the limiting assumption that all involved Hilbert spaces are finite-dimensional, an expression which generates all subspace preserving gluings of a given pair of CPMs, is derived. The nature of the non-uniqueness of gluings and its relation to a proposed definition of subspace locality, is discussed.