Coexistence of qubit effects

We characterize all coexistent pairs of qubit effects. This gives an exhaustive description of all pairs of events allowed, in principle, to occur in a single qubit measurement. The characterization consists of three disjoint conditions, which are easy to check for a given pair of effects. Known special cases are shown to follow from our general characterization theorem.

Figure 1

(Color online) Examples of the specification of effects $\mathit{B}$, which coexist with a given effect $\mathit{A}$. On each figure, parameters $\alpha$, $\mathit{a}$, and $\beta$ are fixed, while the $\mathbf{b}$ vector components $b_{\parallel }$ and $b_{\perp }$ are on $\mathit{x}$ and $\mathit{y}$ axis, respectively. The thick red line denotes the boundary—$\mathit{A}$ and $\mathit{B}$ coexist if and only if the vector $\mathbf{b}$ is inside the shaded region. The thin black circle represents the condition on $\mathit{B}$ to be an effect $b\le \beta$. The blue vector represents the shortest vector $\mathbf{b}$ lying on the boundary—its projection to the $\mathit{x}$ axis equals ${\mathit{b}}_0$. The interval $\left[b_0-w,b_0+w\right]$, where a nontrivial restriction on the length of the allowed vectors $\mathbf{b}$ exists, is denoted by the vertical dot-dashed lines and a black arrow on the $\mathit{x}$ axis. The following parameters are used in the pictures: $\alpha =0.6$ in every figure and (a) $a=0.5$, $\beta =0.6$; (b) $a=0.5$, $\beta =0.9$; (c) $a=0.5$, $\beta =1$; and (d) $a=0.6$, $\beta =0.9$.

Figure 2

The effects $\mathit{A}$ and $\mathit{B}$ are coexistent if and only if the intersection of the four circles is nonempty. The four circles are centered at the corners of a parallelogram with sides $\mathbf{a}$ and $\mathbf{b}$. The circles’ radii are given in the figure and depend on $\gamma$. For the particular $\gamma$ used in the figure, the intersection is empty. Later we will also need the common points of circles 1 and 2 denoted by ${\mathbf{p}}_{\mathbf{1}}$ and ${\mathbf{p}}_{\mathbf{2}}$, and circles 3 and 4 denoted by ${\mathbf{p}}_{\mathbf{3}}$ and ${\mathbf{p}}_{\mathbf{4}}$.

Figure 3

Three different cases where the intersection is one single point. (a) Two circles touch in a single point, which is inside the remaining two circles. (b) Three circles intersect in a single point; it is not the case (a), and the intersection is inside the fourth circle. (c) Four circles intersect in a single point and it is not the case (a) or (b).