Postprocessing of quantum instruments

Studying sequential measurements is of the utmost importanceto both the foundational aspects of quantum theory and the practicalimplementations of quantum technologies, with both of these applications beingabstractly described by the concatenation of quantum instruments into asequence of certain length. In general, the choice of instrument at any givenstep in the sequence can be conditionally chosen based on the classical resultsof all preceding instruments. For two instruments in a sequence we consider theconditional second instrument as an effective way of post-processing the firstinstrument into a new one. This is similar to how a measurement described by apositive operator-valued measure (POVM) can be post-processed into another byway of classical randomization of its outcomes using a stochastic matrix. Inthis work we study the post-processing relation of instruments and the partialorder it induces on their equivalence classes. We characterize the greatest andthe least element of this order, give examples of post-processings betweendifferent types of instruments and draw connections between post-processings ofsome of these instruments and their induced POVMs.