Additionally it is always assumed that measurement outcomes and physical operations are affine maps, i.e. that if Φ Φ is a physical transformation, then we must have Φ Φ ( p x + ( 1 − − p ) y ) = p Φ Φ ( x ) + ( 1 − − p ) Φ Φ ( y ) and similarly for measurement outcomes. This follows from the argument that we should obtain the same outcome if we first prepare a statistical mixture and then apply the physical operation, or if we prepare a statistical mixture of the outcomes of the physical operations.