Extracting remaining information from an inconclusive result in optimal unambiguous state discrimination

Abstract

In unambiguous state discrimination, the measurement results consist of the error-free results and an inconclusive result, and an inconclusive result is conventionally regarded as a useless remainder from which no information about initial states is extracted. In this paper, we investigate the problem of extracting remaining information from an inconclusive result, provided that the optimal total success probability is determined. We present three simple examples. An inconclusive answer in the first two examples can be extracted partial information, while an inconclusive answer in the third one cannot be. The initial states in the third example are defined as the highly symmetric states.

Appendix: Derivation of three linearly independent symmetric states

The input states we explore are given in Ref. [38]. The authors defined the symmetric states

$$\begin{aligned}&\left| {\varphi _1 } \right\rangle =c_1 \left| 1 \right\rangle +c_2 \left| 2 \right\rangle +c_3 \left| 3 \right\rangle ,\nonumber \\&\left| {\varphi _2 } \right\rangle =c_1 \left| 1 \right\rangle +e^{i{2\pi }/3}c_2 \left| 2 \right\rangle +e^{-i{2\pi }/3}c_3 \left| 3 \right\rangle ,\nonumber \\&\left| {\varphi _3 } \right\rangle =c_1 \left| 1 \right\rangle +e^{-i{2\pi }/3}c_2 \left| 2 \right\rangle +e^{i{2\pi }/3}c_3 \left| 3 \right\rangle , \end{aligned}$$

(41)

where the arbitrary $c_k$ coefficients satisfy the normalization condition $\sum _{i=1}^3 {\left| {c_i } \right| ^{2}} =1$. The complex inner product $S=\left| S \right| e^{i\theta }$ among the symmetric states is defined by

$$\begin{aligned} S=\left\langle {\varphi _1 } \right. \left| {\varphi _2 } \right\rangle =\left\langle {\varphi _2 } \right. \left| {\varphi _3 } \right\rangle =\left\langle {\varphi _3 } \right. \left| {\varphi _1 } \right\rangle =\left| S \right| e^{i\theta }. \end{aligned}$$

(42)

The symmetric states are linearly independent if and only if $S\in \left[ {0,\left| {S_\theta } \right| } \right)$ [38], where

$$\begin{aligned} \left| {S_\theta } \right| =\left\{ {\begin{array}{l} \frac{-1}{2\cos \theta },\quad \quad \quad \quad \quad \theta \in \left[ {{2\pi }/3,{4\pi }/3} \right] \\ \frac{-1}{2\cos \left( {\theta +{2\pi }/3} \right) },\quad \quad \theta \in \left[ {0,{2\pi }/3} \right] \\ \frac{-1}{2\cos \left( {\theta -{2\pi }/3} \right) },\quad \quad \theta \in \left[ {{4\pi }/3,2\pi } \right] \\ \end{array}} \right. , \end{aligned}$$

(43)

Note that $S=0$ implies the states are orthogonal and $S=\left| {S_\theta } \right|$ implies three linearly dependent states. In our paper, we take $S\in \left( {0,\left| {S_\theta } \right| } \right)$, implying three linearly independent symmetric states. We use the second expression in Eq. (43). For $\beta \in \left( {0,\pi /4} \right)$, we define the phase factor, $\theta$, as

$$\begin{aligned} \tan \theta =\frac{\frac{\sqrt{3}}{2}\sin ^{2}\beta }{1-\frac{3}{2}\sin ^{2}\beta }\in \left( {0,\sqrt{3}} \right) , \quad \theta \in \left( {0,\pi /3} \right) . \end{aligned}$$

(44)

So the second expression turns to

$$\begin{aligned} \left| {S_\theta } \right| =\frac{-1}{2\cos \left( {\theta +{2\pi }/3} \right) }=\sqrt{1-3\sin ^{2}\beta \cos ^{2}\beta }\in \left( {\frac{1}{2},1} \right) \!. \end{aligned}$$

(45)

When $S=\left| {S_\theta } \right|$, the symmetric states become linearly dependent. For example, the linearly dependent symmetric states are given by Eq. (59) [29]

$$\begin{aligned}&\left| {\tilde{\varphi }_1 } \right\rangle =\cos \beta \left| {\bar{{1}}} \right\rangle +\sin \beta \left| {\bar{{2}}} \right\rangle \!,\nonumber \\&\left| {\tilde{\varphi }_2 } \right\rangle =\cos \beta \left| {\bar{{1}}} \right\rangle +e^{i\frac{2\pi }{3}}\sin \beta \left| {\bar{{2}}} \right\rangle \!,\nonumber \\&\left| {\tilde{\varphi }_3 } \right\rangle =\cos \beta \left| {\bar{{1}}} \right\rangle +e^{i\frac{4\pi }{3}}\sin \beta \left| {\bar{{2}}} \right\rangle \!, \end{aligned}$$

(46)

with the complex inner product

$$\begin{aligned} S=\left\langle {\tilde{\varphi }_1 } \right. \left| {\tilde{\varphi }_2 } \right\rangle =\left\langle {\tilde{\varphi }_2 } \right. \left| {\tilde{\varphi }_3 } \right\rangle =\left\langle {\tilde{\varphi }_3 } \right. \left| {\tilde{\varphi }_1 } \right\rangle \nonumber \\ =\cos ^{2}\beta +e^{i\frac{2\pi }{3}}\sin ^{2}\beta =\left| {S_\theta } \right| e^{i\theta }. \end{aligned}$$

(47)