Quantum effects, sequential independence and majorization

A quantum effect is a positive Hilbert space contraction operator. If {Ei}, 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 {Ei}, 1⩽i⩽n, are sequentially independent if and only if for every permutation r1r2…rn of the set Sn={1,2,…,n}. The sequential independence of the effects Ei, 1⩽i⩽n, implies EnoEn-1o…oEj+1oEjo…oE1=(EnoEn-1o…Ej+1)oEjo…oE1 for every 1⩽j⩽n. It is proved that if there exists an effect Ej, 1⩽j⩽n, such that Ej⩽(EnoEn-1o…Ej+1)oEjo…oE1, then the effects {Ei} are sequentially independent and satisfy .