Lemma 5 ([Transporting answer measurements]). Let H, K be finite-dimensional complex Hilbert spaces, let T be an effect on H , and let F: H→ K satisfy F^*F=T . Suppose that A is a nonempty finite set and that positive operators (G_a)_a∈ A …

All inverses below act on the support of T and are zero on its kernel. Define V_T=FT^ -1/2 . Then V_T^*V_T=Π_T , the orthogonal projection onto suppT , so V_TV_T^* is an orthogonal projection on K . Each G_a is supported on suppT , because 0 G_a T . Fix a_*∈ A and put G_a =V_TT^ -1/2 G_aT^ -1/2 V_T^* +1_ a=a_* (I_ K -V_TV_T^*). These operators are positive and sum to I_ K . Since F=V_TT^ 1/2 , their pullbacks are G_a , as claimed. The same formula applies when T=0 : all G_a and V_T then vanish, and the POVM always outputs a_* .