Coexistence of Qubit Effects
Coexistence of Qubit Effects
Suppose two qubit measurements are given by the effects and , which are parametrized by values of bias and and sharpness and , which are the lengths of the corresponding vectors and that give the directions along which the measurements are performed. The projection of onto is labeled as ( axis) and the orthogonal complement is labeled as ( axis). For given , and , the graph shows the area for possible choices of such for which the effects and are simultaneously measurable, that is, coexistent (the area shaded gray bounded by the thick red curve). The thin black circle represents the condition for to be an effect (a valid measurement). The dot-dashed lines (if present) define the region within which there is a nontrivial restriction on the vector , in which case either an orange or a red light shines, as opposed to a green light, which corresponds to unrestricted joint measurability. A dashed line, together with the blue vector, shows the strictest limitation on for the chosen parameters—for all such that they are not longer than the blue vector, there is no angle restriction.
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