Figure 2. Information geometry of qubit observables.

The largest green cone represents the complementarity chamber at the completely mixed state (cf. Fig. 1). The two upward red cones represent the sets of hypothetical Fisher information matrices lower bounded by the Fisher information matrices of two sharp von Neumann observables (corresponding to the tips of the cones), respectively. The two observables are incompatible since the intersection of the two cones is disjoint from the complementarity chamber. The distance from the intersection to the base of the complementarity chamber quantifies the degree of incompatibility. By contrast, their noisy versions corresponding to the tips of the two smaller green cones are compatible.