  We present a barycentric decomposition for quantum instruments whose output
space is finite-dimensional and input space is separable. As a special case, we
obtain a barycentric decomposition for channels between such spaces and for
normalized positive-operator-valued measures in separable Hilbert spaces. This
extends the known results by Ali and Chiribella et al. on decompositions of
quantum measurements, and formalises the fact that every instrument between
finite-dimensional Hilbert spaces can be represented using only finite-outcome
instruments.
