Quantum effects, sequential independence and majorization

A quantum effect is a positive Hilbert space contraction operator. If {E_i}, 1?i?n, are n quantum effects (defined on some Hilbert space H), then their sequential product is the operator . It is proved that the quantum effects {E_i}, 1?i?n, are sequentially independent if and only if for every permutation r₁r₂…r_n of the set S_n={1,2,…,n}. The sequential independence of the effects E_i, 1?i?n, implies E_noE_n-1o…oE_j+1oE_jo…oE₁=(E_noE_n-1o…E_j+1)oE_jo…oE₁ for every 1?j?n. It is proved that if there exists an effect E_j, 1?j?n, such that E_j?(E_noE_n-1o…E_j+1)oE_jo…oE₁, then the effects {E_i} are sequentially independent and satisfy .