Finding optimal measurements with inconclusive results using the problem of minimum error discrimination

We propose an approach for finding an optimal measurement for quantum state discrimination that maximizes the probability of correct detection with a fixed rate of inconclusive results. In our approach, we obtain the optimal measurement by solving the problem of finding a measurement that maximizes the weighted sum of the probability of correct detection and that of inconclusive results. We show that this problem can be reduced to the widely studied problem of finding a minimum error measurement for a certain state set, which maximizes the probability of correct detection without inconclusive results. As an application of our approach, we show how to solve the problem of finding an optimal measurement for qubit states with a fixed rate of inconclusive results.

Figure 1

Typical behavior of $P_{\mathrm{C}}^{\circ }\left(p\right)$ and $F\left(\alpha \right)$.

Figure 2

Example of a geometric representation for the problem of finding an MOIM for a three qubit state set $\rho =\{{\widehat{\rho }}_m:m\in {\mathcal{I}}_3\}$ (projected on the two-dimensional plane that includes the centers of three balls ${\mathcal{B}}_0,{\mathcal{B}}_1$, and ${\mathcal{B}}_2)$. ${\mathcal{B}}_m(m\in {\mathcal{I}}_3)$ (in black), ${\mathcal{B}}_3$ (blue), and ${\mathcal{B}}_Z$ (red, dashed) respectively correspond to ${\widehat{\rho }}_m,\alpha \widehat{G}$, and $\widehat{Z}$. (a) ${\mathcal{B}}_3$ is not tangent to ${\mathcal{B}}_Z$. (b) ${\mathcal{B}}_3$ is tangent to ${\mathcal{B}}_Z$ and ${\mathcal{B}}_3\ne {\mathcal{B}}_Z$. (c) ${\mathcal{B}}_3={\mathcal{B}}_Z$.

Figure 3

Bloch sphere representation of a three mirror symmetric state set in the cross section of the two-dimensional plane $x=y=0$ (in the case of $X_z\ge G_z{X}_t)$.

Figure 4

Bloch sphere representation of a three mirror symmetric state set in the cross section of the plane $(y,t)=(0,{\xi }_{min})$ (in the case of $X_z<G_z{X}_t)$.

Figure 5

$P_{\mathrm{C}}^{\circ }\left(p\right)$ and $F\left(\alpha \right)$ for a three mirror symmetric state set with $\xi =1/3,{\eta }_0={\eta }_1=0$, and $\theta =0.26\pi$, which satisfies $X_z<G_z{X}_t$.