Optimal bounded-error strategies for projective measurements in nonorthogonal-state discrimination

Research in nonorthogonal-state discrimination has given rise to two conventional optimal strategies: unambiguous discrimination (UD) and minimum error discrimination. We explore the experimentally relevant range of measurement strategies between the two, where the rate of inconclusive results is minimized for a bounded-error rate. We first provide some constraints on the problem that apply to generalized measurements [positive-operator-valued measurements (POVMs)]. We then provide the theory for the optimal projective measurement in this range. Through analytical and numerical results we investigate this family of projective, bounded-error strategies and compare it to the POVM family as well as to experimental implementation of UD using POVMs. We also discuss a possible application of these bounded-error strategies to quantum key distribution.

Figure 1

Bounds on the minimum inconclusive rate function ${\tilde{P}}_{\mathrm{In}}\left(ϵ\right)$ for the $n$-state problem with equal prior probabilities. The dash-dotted line with a slope of $-\frac{n}{n-1}$ starting at $(0,P_{\mathrm{UD}})$ represents strategies where random guesses are made when inconclusive outcomes are obtained from the UD strategy. The dashed line that connects the point (0,1) to $(P_{\mathrm{ME}},0)$ represents strategies where results for error-prone outcomes from the ME strategy are interpreted as inconclusive. Similar strategy manipulations (shown as dotted lines) apply to any existing strategy, represented by $(P_{{\mathrm{E}}_0},P_{{\mathrm{In}}_0})$, and so provide two constraints on the minimizing function ${\tilde{P}}_{\mathrm{In}}\left(ϵ\right)$: a bound on its slope (it must be less than $-\frac{n}{n-1}$) and a bound on the neighboring points of the function. ${\tilde{P}}_{\mathrm{In}}\left(ϵ\right)$ then lies somewhere in the shaded region.

Figure 2

The geometry of the real, two-state problem. The input states are represented by $\mid {\psi }_1⟩$ and $\mid {\psi }_2⟩$, and the PVM elements project onto $p_1$ and $p_2$. The angle $\varphi$ offsets the orientation of the PVM from the zero-error UD case where $p_1$ is interpreted as $\mid {\psi }_1⟩$ and $p_2$ is interpreted as inconclusive. $\theta$ is the separation angle between the states.

Figure 3

(Color online) The two-state discrimination problem for $\theta =\pi ∕4$ and ${\eta }_1={\eta }_2=1∕2$. As $P_{\mathrm{E}}\to 0$, the optimal PVM curve (solid line) approaches the dashed curve representing the less general optimization over the orientation of the PVM with $w_2=0$. For error approaching the ME value, the optimal PVM curve approaches the dash-dotted line representing strategies based on the ME strategy but with $0<w_2<1$. For any value of $\theta$, the dashed and dash-dotted curves intersect at $(\frac{P_{\mathrm{ME}}}{2},\frac{1}{2})$ where $w_2=0$, shown here as a diamond marker. The optimal POVM curve for the same parameters is shown as the dotted line.

Figure 4

State discrimination for the three states used in Ref. [17], $\mid {\psi }_1⟩=(\sqrt{2∕3},0,1∕\sqrt{3})$, $\mid {\psi }_2⟩=(0,1∕\sqrt{3},\sqrt{2∕3})$, and $\mid {\psi }_3⟩=(0,-1∕\sqrt{3},0,\sqrt{2∕3})$. The optimal projective strategies are represented by the solid line. The two dashed lines ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ represent the strategies that discriminate one and two input states, respectively, using only the orientation of the PVM. The diamond and circular marker represent the minimum error for ${\mathrm{C}}_2$, $P_{\mathrm{E},2}^{\min }$, and the minimum error $(0,P_{\mathrm{ME}})$ respectively. Using the PVM orientation defined at those points, ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ are generated by reducing the weight corresponding to the discriminated state $(w_i=1)$ that is hardest to discriminate. The optimal PVM at 3% error gives a correct rate of 62.3% compared to the 54.5% attained by the experimental POVM in [17] shown here as a square. The optimal POVM is shown as the dotted line.