Sequentially independent effects
Stanley Gudder, Gabriel Nagy · Proceedings of the American Mathematical Society · 2001
A quantum effect is a yes-no measurement that may be unsharp. An effect is represented by an operator E E on a Hilbert space that satisfies 0 ≤ E ≤ I 0\le E\le I . We define effects E 1 , E 2 , … , E n E_1,E_2,\ldots ,E_n to be sequentially independent if the result of any sequential measurement of E 1 , E 2 , … , E n E_1,E_2,\ldots ,E_n does not depend on the order in which they are measured. We show that two effects are sequentially independent if and only if they are compatible. That is, their corresponding operators commute. We also show that three effects are sequentially independent if and only if all permutations of the product of their corresponding operators coincide. It is noted that this last condition does not imply that the three effects are mutually compatible.