Any quantum instrument on a system S can be modeled as a projective measurement on S and (jointly) an uncorrelated auxiliary A followed by a unitary conditional on the measurement outcome. Let η η (with η η > 0 and T r η η = 1 ) be the normalized initial state of A , let { Π Π i } (with Π Π i = Π Π i † † = Π Π i 2 and Π Π i Π Π j = δ δ i j Π Π i ) be a projective measurement on S A , and let { U i } (with U i † † = U i − − 1 ) be unitaries on S A . Then one can check that