# Discrimination of Non-Local Correlations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Nonsignaling Box

## 3. Local Polytope

## 4. Distance from the Local Polytope

**Problem**

**1.**

## 5. Computing the Distance

#### 5.1. Necessary and Sufficient Conditions for Optimality

- First KKT condition (stationarity condition): The gradient of the Lagrangian is equal to zero. The Lagrangian of Problem 1 is$$\mathcal{L}=F\left[\chi \right]-{\displaystyle \sum _{\mathbf{r},\mathbf{s}}}\lambda (\mathbf{r},\mathbf{s})\chi (\mathbf{r},\mathbf{s}),$$
- Second KKT condition (feasibility of the constraints): The function $\chi $ is non-negative, $\chi (\mathbf{r},\mathbf{s})\ge 0$.
- Third condition (dual feasibility): The Lagrange multipliers $\lambda $ are non-negative; that is,$$\lambda (\mathbf{r},\mathbf{s})\ge 0.$$
- Fourth condition (complementary slackness): If $\chi (\mathbf{r},\mathbf{s})\ne 0$, then the multiplier $\lambda (\mathbf{r},\mathbf{s})$ is equal to zero; that is,$$\lambda (\mathbf{r},\mathbf{s})\chi (\mathbf{r},\mathbf{s})=0.$$

**Problem**

**2.**

#### 5.2. Overview of the Algorithm

**Lemma**

**1.**

**Proof.**

#### 5.3. The Algorithm

**Algorithm**

**1.**

- 1.
- Set $({\mathbf{r}}^{\prime},{\mathbf{s}}^{\prime})$ equal to the sequences given by the oracle with $P(r,s|a,b)$ as query.
- 2.
- Set $\mathsf{\Omega}=\left\{({\mathbf{r}}^{\prime},{\mathbf{s}}^{\prime})\right\}$.
- 3.
- Compute the optimizers $\chi (\mathbf{r},\mathbf{s})$ and $\rho (r,s|a,b)$ of Problem 2. The associated F provides an upper bound of the optimal value ${F}^{min}$.
- 4.
- Consult the oracle with $g(r,s;a,b)=P(r,s|a,b)-\rho (r,s|a,b)$ as query. Set $({\mathbf{r}}^{\prime},{\mathbf{s}}^{\prime})$ and α are equal to the sequences returned by the oracle and the associated maximal value, respectively. That is,$$({\mathbf{r}}^{\prime},{\mathbf{s}}^{\prime})={\displaystyle \underset{(\mathbf{r},\mathbf{s})}{\mathit{argmax}}}{\displaystyle \sum _{a,b}}g({r}_{a},{s}_{b};a,b)W(a,b),\phantom{\rule{-14.22636pt}{0ex}}$$$$\alpha ={\displaystyle \sum _{a,b}}g({r}_{a}^{\prime},{s}_{b}^{\prime}|a,b)W(a,b),$$
- 5.
- Compute a lower bound on the ${F}^{min}$ from ρ and α (see following discussion and Section 6.1). The difference between the upper and lower bounds provides an upper bound on the reached accuracy.
- 6.
- If a given accuracy is reached, stop.
- 7.
- Remove from Ω the points where χ is zero and add $({\mathbf{r}}^{\prime},{\mathbf{s}}^{\prime})$.
- 8.
- Go back to Step 3.

#### 5.4. Stopping Criterion for Algorithm 1

#### 5.5. Stopping Criterion for Problem 2 (Optimization at Step 3 of Algorithm 1)

- The sequence ${F}_{0}^{min},{F}_{1}^{min},\cdots $ of the exact optimal values of Problem 2, with $\mathsf{\Omega}={\mathsf{\Omega}}_{0},{\mathsf{\Omega}}_{1},\cdots $, is monotonically decreasing.
- The sets ${\mathsf{\Omega}}_{0},{\mathsf{\Omega}}_{1},\cdots $ contain points associated with linearly independent vertices of the local polytope, implying that the cardinality of ${\mathsf{\Omega}}_{n}$ is never greater than ${d}_{NS}+1$.

#### 5.6. Cleaning Up (Step 7)

**Theorem**

**1.**

**Corollary**

**1.**

#### 5.7. Solving Problem 2

## 6. Convergence Analysis and Computational Cost

#### 6.1. Dual Problem

**Problem**

**3**(dual problem of Problem 1).

**Problem**

**4**(dual problem of Problem 2).

#### 6.2. Convergence and Polynomial Cost

#### 6.3. Simulation of the Oracle

**Algorithm**

**2.**

- 1.
- Generate a random sequence $\mathbf{r}$.
- 2.
- Maximize ${\sum}_{a,b}}g({r}_{a},{s}_{b};a,b)W(a,b)$ with respect to the sequence $\mathbf{s}$ (see later discussion).
- 3.
- Maximize ${\sum}_{a,b}}g({r}_{a},{s}_{b};a,b)W(a,b)$ with respect to the sequence $\mathbf{r}$.
- 4.
- Repeat from Step 2 until the block-maximizations stop making progress.

## 7. Numerical Tests

#### 7.1. Maximally Entangled State

#### 7.2. Non-Maximally Entangled State

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Time required for computing the distance from the local polytope for a maximally entangled state as a function of the number of measurements (log-log scale) with accuracy equal to ${10}^{-3}$, ${10}^{-4}$, and ${10}^{-5}$ (red, blue, and green points, respectively).

**Figure 2.**Distance from the local polytope as a function of $\gamma $ in the unbiased case (red stars) and biased case (blue triangles).

**Figure 3.**Time required for computing the distance from the local polytope as a function of the number of measurements (log-log scale) in the unbiased case, for $\gamma =0.8$ (red stars) and $\gamma =0.6$ (blue triangles). The green lines are the functions ${10}^{-6}{M}^{6}$ and ${10}^{-8}{M}^{6}$. The accuracy is ${10}^{-5}$.

**Figure 4.**The same as Figure 3 in the biased case, for $\gamma =0.8$ (red stars), $\gamma =0.6$ (blue triangles), and $\gamma =0.4$ (green circles).

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**MDPI and ACS Style**

Montina, A.; Wolf, S.
Discrimination of Non-Local Correlations. *Entropy* **2019**, *21*, 104.
https://doi.org/10.3390/e21020104

**AMA Style**

Montina A, Wolf S.
Discrimination of Non-Local Correlations. *Entropy*. 2019; 21(2):104.
https://doi.org/10.3390/e21020104

**Chicago/Turabian Style**

Montina, Alberto, and Stefan Wolf.
2019. "Discrimination of Non-Local Correlations" *Entropy* 21, no. 2: 104.
https://doi.org/10.3390/e21020104