Approximating incompatible von Neumann measurements simultaneously

We study the problem of performing orthogonal qubit measurements simultaneously. Since these measurements are incompatible, one has to accept additional imprecision. An optimal joint measurement is the one with the least possible imprecision. All earlier considerations of this problem have concerned only joint measurability of observables, while in this work we also take into account conditional state transformations (i.e., instruments). We characterize the optimal joint instrument for two orthogonal von Neumann instruments as being the Lüders instrument of the optimal joint observable.

Figure 1

(a) An observable describes the measurement outcome statistics, whereas (b) an instrument describes both the measurement outcome statistics and the conditional output states with possible separation in accordance with the classical outcomes.

Figure 2

Visual representation of the four-ball intersection determining the existence of joint observable for $\mathsf{A}$ and $\mathsf{B}$ parametrized by $\gamma$ and vector $\mathbf{g}$. If $\gamma =1/2$, all four balls intersect in a single point (white circle) defining the joint observable (solid arcs). If $\gamma$ differs from $1/2$ (e.g., $\gamma =0.4$ as in the figure), two opposite circles intersect, while the other two do not; in such a case there is no joint observable with given $\gamma$ (dot-dashed arcs).

Figure 3

Bloch vector representation of instrument $\mathcal{G}$ gives us in the case of approximation ${\mathcal{B}}_{+}$ the resulting normalized output states lying on a (thick horizontal) blue line. The average distance is minimized for the choice of symmetrical pure states ${\xi }_{\pm \pm }$. In the case of approximation ${\mathcal{A}}_{+}$ the normalized output states are represented by (thick vertical) red line with red dots representing the worst cases. These states are closest to $\mathbf{x}$ when the output state is noisier than in the average case.

Figure 4

Bloch vector comparison of the four- and eight-outcome instruments. The four-outcome instrument leads to normalized output states from the edges of the tetrahedron (especially for the measurement along the $x$ axis illustrated by the thick red diagonal). The eight-outcome instrument leads to the faces of the cube given by the vectors ${\mathbf{q}}_{\pm \pm \pm }$ (especially for the measurement along the $x$ axis illustrated by the blue circle).