# Structure of Optimal State Discrimination in Generalized Probabilistic Theories

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## Abstract

**:**

## 1. Introduction

## 2. Optimal State Discrimination in GPTs

#### 2.1. Generalized Probabilistic Theories

#### 2.2. State Discrimination in Convex Optimization

#### A Convex Optimization Framework

#### 2.3. Constraint Qualification

#### 2.4. The Complementarity Problem

- The first condition, symmetry parameter, follows from the Lagrangian stability and shows that for any discrimination problem e.g., ${\{{q}_{\mathrm{x}},{w}_{\mathrm{x}}\}}_{\mathrm{x}=1}^{N}$, there exists a single parameter K which is decomposed into N different ways with given states and complementary states ${\{{r}_{\mathrm{x}},{d}_{\mathrm{x}}\}}_{\mathrm{x}=1}^{N}$. Then, the second condition in Equation (7) from the complementary slackness characterizes optimal effects by the orthogonality relation between complementary states and optimal effects. These generalize optimality conditions from quantum cases to all GPTs, see also various forms of optimality conditions in quantum cases [14].
- Primal and dual parameters satisfying the KKT conditions are automatically optimal parameters that provide solutions in the primal and the dual problems. Note also that, since the strong duality holds, both problems show the same solution. Conversely, the fact that the strong duality holds in Equations (4) and (5) implies the existence of optimal parameters which satisfy KKT conditions and give the guessing probability in Equation (1).

#### 2.5. The Geometric Method and the General Form of the Guessing Probability

## 3. Examples: Polygon States

#### 3.1. A Case of $N=3$

#### 3.2. A Case of $N=4$

- (i) ${\{{f}_{\mathrm{x}}/2\}}_{\mathrm{x}=0}^{3}$: In this case, we have$$\begin{array}{c}\hfill p\left(\mathrm{x}\right|\mathrm{x})=\frac{1}{2}{f}_{\mathrm{x}}\left[{w}_{\mathrm{x}}\right]=\frac{1}{2},\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}\mathrm{thus},\phantom{\rule{3.33333pt}{0ex}}{p}_{\mathrm{guess}}=\frac{1}{4}\sum _{x=0}^{3}p\left(\mathrm{x}\right|\mathrm{x})=\frac{1}{2}.\end{array}$$
- (ii) $\{{f}_{0},{f}_{2}\}$: In this case, measurement on effect on ${e}_{0}$ concludes that given state is either ${w}_{0}$ or ${w}_{3}$, and ${e}_{3}$ to ${w}_{1}$ or ${w}_{2}$. This is because, from the orhogonality condition in Equation (7), it holds that$$\begin{array}{c}\hfill {f}_{0}\left[{d}_{0}\right]={f}_{0}\left[{w}_{2}\right]=0,\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}{f}_{0}\left[{d}_{3}\right]={e}_{0}\left[{w}_{1}\right]=0.\end{array}$$
- (iii) $\{{f}_{1},{f}_{3}\}$: This case works in a similar way to the previous. Measurement on effect on ${f}_{1}$ concludes that given state is either ${w}_{0}$ or ${w}_{1}$, and ${f}_{3}$ to ${w}_{2}$ or ${w}_{3}$.

#### 3.3. When No Measurement Is Optimal

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Bae, J.; Kim, D.-G.; Kwek, L.-C.
Structure of Optimal State Discrimination in Generalized Probabilistic Theories. *Entropy* **2016**, *18*, 39.
https://doi.org/10.3390/e18020039

**AMA Style**

Bae J, Kim D-G, Kwek L-C.
Structure of Optimal State Discrimination in Generalized Probabilistic Theories. *Entropy*. 2016; 18(2):39.
https://doi.org/10.3390/e18020039

**Chicago/Turabian Style**

Bae, Joonwoo, Dai-Gyoung Kim, and Leong-Chuan Kwek.
2016. "Structure of Optimal State Discrimination in Generalized Probabilistic Theories" *Entropy* 18, no. 2: 39.
https://doi.org/10.3390/e18020039