Preprocessing operations and the reverse compression

Abstract: The task of compression of data -- as stated by the source coding theorem -- is one of the cornerstones of information theory. Data compression usually exploits statistical redundancies in the data according to its prior distribution. Motivated by situations where one does not have access to the statistics of data, but has some information about a transformation that is going to be applied to it, we propose a novel method for data compression called reverse compression. It is defined in such a way that works for both classical and quantum information processes, and furthermore relies exclusively on the channel to be used: all input data leading to indistinguishable outputs is compressed to the same state, regardless of their prior distribution. Moreover, this process can be characterized as a higher order operation within the type of preprocessing. We also consider as an example the application of the method to the classical and quantum erasure channel. The examples suggest that noiseless reverse compression can take place only in trivial cases, although meaningful instances of noisy reverse compression can exist.