Sequentially independent effects

A quantum effect is a yes-no measurement that may be unsharp. An effect is represented by an operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a Hilbert space that satisfies <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 less-than-or-equal-to upper E less-than-or-equal-to upper I"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo> ≤ </mml:mo> <mml:mi>E</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mi>I</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">0\le E\le I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We define effects <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E 1 comma upper E 2 comma ellipsis comma upper E Subscript n Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">E_1,E_2,\ldots ,E_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to be sequentially independent if the result of any sequential measurement of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E 1 comma upper E 2 comma ellipsis comma upper E Subscript n Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">E_1,E_2,\ldots ,E_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.