Discriminating two nonorthogonal states against a noise channel by feed-forward control

We propose a scheme by using the feed-forward control (FFC) to realize a better effect of discrimination of two nonorthogonal states after passing a noise channel based on the minimum-error (ME) discrimination. We show that the application of our scheme can highly improve the effect of discrimination compared with the ME discrimination without the FFC for any pair of nonorthogonal states and any degree of amplitude damping. Especially, the effect of our optimal discrimination can reach that of the two initial nonorthogonal pure states in the presence of the noise channel in a deterministic way for equal a priori probabilities or even be better than that in a probabilistic way for unequal a priori probabilities.

Figure 1

Schematic diagram showing the procedure of discrimination between two nonorthogonal mixed states in quantum communication. (a) The ME discrimination without the FFC (conventional scheme). (b) The ME discrimination based on the FFC (our scheme).

Figure 2

Maximum correct probabilities $P_C^{\mathrm{fin}}$ [Eq. (36)] (solid-blue lines) and $P_C^{\mathrm{mix}}$ [Eq. (13)] (dashed-olive lines) as functions of the measurement strength $p$ for (a) $\theta =\pi /16$, (b) $\theta =4\pi /16$, and (c) $\theta =6\pi /16$. Panel (d) plots the success probability $P_S$ corresponding to panels (a)–(c) for our scheme which is state independent. $r=0.7;q_{+}=q_{-}=1/2$.

Figure 3

Maximum correct probabilities $P_C^{\mathrm{fin}}$ [Eq. (36)] (solid-blue lines) and $P_C^{\mathrm{mix}}$ [Eq. (13)] (dashed-olive lines) as functions of the measurement strength $p$ for (a) $r=0.1$, (b) $r=0.5$, and (c) $r=0.9$. The insets plot the corresponding success probabilities $P_S$ for our scheme. $\theta =6\pi /16;q_{+}=q_{-}=1/2$.

Figure 4

Maximum correct probabilities $P_C^{\mathrm{fin}}$ [Eq. (30)] (solid-blue lines) and $P_C^{\mathrm{mix}}$ [Eq. (12)] (dashed-olive lines) as functions of the measurement strength $p$ for (a) $\theta =\pi /16$, (b) $\theta =4\pi /16$, and (c) $\theta =6\pi /16$. Panel (d) plots the success probability $P_S$ corresponding to panels (a)–(c) for our scheme which is state independent. $r=0.7;q_{+}=1/3;q_{-}=2/3$.

Figure 5

Maximum correct probabilities $P_C^{\mathrm{fin}}$ [Eq. (30)] (solid-blue lines) and $P_C^{\mathrm{mix}}$ [Eq. (12)] (dashed-olive lines) as functions of the measurement strength $p$ for (a) $r=0.1$, (b) $r=0.5$, and (c) $r=0.9$. The insets plot the corresponding success probabilities $P_S$ for our scheme. $\theta =6\pi /16;q_{+}=1/3;q_{-}=2/3$.

Figure 6

$P_C^{\mathrm{fin}}-P_S$ diagrams for our scheme (blue region) with $P_C^{\mathrm{mix}}$ [Eq. (13)] (dotted-olive lines) and $P_C^{\mathrm{pure}}$ [Eq. (7)] (dashed-red lines) for (a) $\theta =\pi /16$, (b) $\theta =4\pi /16$, and (c) $\theta =6\pi /16$. $r=0.7;q_{+}=q_{-}=1/2$. The blue lines represent our optimal lines which coincide with $P_C^{\mathrm{pure}}$ and the purple circles represent our optimal points.

Figure 7

$P_C^{\mathrm{fin}}-P_S$ diagram for different degrees of amplitude damping. The solid-blue line represents both our scheme for $r=0.1$, $r=0.5$, and $r=0.9$ and $P_C^{\mathrm{pure}}$ [Eq. (7)] which coincide; $P_C^{\mathrm{mix}}$ [Eq. (13)] is represented by dotted-olive line ($r=0.1$), dashed-olive line ($r=0.5$), and dashed dotted-olive line ($r=0.9$). $\theta =6\pi /16;q_{+}=q_{-}=1/2$.

Figure 8

$P_C^{\mathrm{fin}}-P_S$ diagrams for our scheme (solid-blue lines) with $P_C^{\mathrm{mix}}$ [Eq. (12)] (dotted-olive lines) and $P_C^{\mathrm{pure}}$ [Eq. (6)] (dashed-red lines) for (a) $\theta =\pi /16$, (b) $\theta =4\pi /16$, and (c) $\theta =6\pi /16$. $r=0.7;q_{+}=1/3;q_{-}=2/3$. The purple circles represent our optimal points.

Figure 9

$P_C^{\mathrm{fin}}-P_S$ diagram for different degrees of amplitude damping. The solid-blue line represents our scheme for $r=0.1$, $r=0.5$, and $r=0.9$ which coincide. The dashed-red line represents $P_C^{\mathrm{pure}}$ [Eq. (6)]; $P_C^{\mathrm{mix}}$ [Eq. (12)] is represented by dotted-olive line ($r=0.1$) and dashed dotted-olive line ($r=0.5$ and $r=0.9$ which coincide). $\theta =6\pi /16;q_{+}=1/3;q_{-}=2/3$.