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Experimental Connection between the Instrumental and Bell Inequalities^{ †}

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## Abstract

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## 1. Introduction

## 2. Results

#### 2.1. Active Feed-Forward of Information

#### 2.2. Post-Selection Scenario

#### 2.2.1. Relation between Bonet and CHSH Inequality Violation

#### 2.2.2. CHSH Violation within an Instrumental Process

## 3. Discussion

## 4. Materials and Methods

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CHSH | Clauser-Horne-Shimony-Holt inequality |

DAG | Directed Acyclic Graph |

HWP | Half-wave plate |

PBS | Polarizing Beam Splitter |

BBO | Beta-Barium Borate |

SPDC | Spontaneous Parametric Down-Conversion |

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**Figure 1.**

**Directed Acyclic Graphs representing causal models.**In this figure, we show three different causal models represented by Directed Acyclic Graphs (DAG) [2], where each node represents a variable and the arrows link variables between whom there is a causal relationship. (

**a**) This is the simplest causal model in which a variable A has an influence over B, but it has no testable mathematical constraints characterizing the allowed joint probabilities $p(a,b)$. (

**b**) The causal model of a CHSH scenario, where the parties Alice (A) and Bob (B) share a system and choose the basis on which to measure it according two independent variables, X and Y. Hence, no communication must occur between A and B. (

**c**) DAG representing the instrumental scenario, whose main difference from the CHSH Bell-like scenario, is the presence of a classical channel of communication between the parties, i.e., A has a causal influence over B.

**Figure 2.**

**Experimental Apparatus.**We generate a singlet state through a SPDC process (constituting our hidden variable $\Lambda $) that is shared by the two parties A and B. On A’s path, the measurement station is made by a HWP followed by a PBS, since the qubits are encoded in the polarization of the photons. On B’s path, there is an analogous measurement station, but, before the HWP ${H}_{2}$, we put a Pockels cell, to switch from one operator to the other, in the case of active feed-forward, i.e., triggering a high voltage application to the Pockels cell and making it behave as a HWP, while, when it is not triggered, it performs the identity. The voltage application is triggered by Alice’s detector ${A}_{1}$ (i.e., corresponding to output 0), whose signal is split and sent both to a coincidence counter and to the Pockels cell. In the post-selection case, the Pockels cell is not triggered and A and B choose independently the measurement basis, performing the desired observables rotating respectively ${H}_{1}$ and ${H}_{2}$.

**Figure 3.**

**Experimental results.**In this plot we show the experimental quantum violations of Bonet and CHSH inequalities obtained from 16 experimental runs in the post-selection regime (for both scenarios the classical upper bound is 2). In purple and blue, we show respectively the Bonet’s and CHSH violations obtained through the apparatus described in Figure 2 section, keeping the Pockels cell switched off and performing Alice’s and Bob’s measurements, rotating the HWPs. In orange, we show the extent of the Bonet inequality’s violation that can be obtained from the probabilities p(a,b|x,y) belonging to the CHSH scenario. In green, we show the CHSH violation obtained on an instrumental process platform, through an alternative procedure which does not require the correlations between all of the 4 combinations of the inputs (x,y).

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**MDPI and ACS Style**

Agresti, I.; Carvacho, G.; Poderini, D.; Aolita, L.; Chaves, R.; Sciarrino, F.
Experimental Connection between the Instrumental and Bell Inequalities. *Proceedings* **2019**, *12*, 27.
https://doi.org/10.3390/proceedings2019012027

**AMA Style**

Agresti I, Carvacho G, Poderini D, Aolita L, Chaves R, Sciarrino F.
Experimental Connection between the Instrumental and Bell Inequalities. *Proceedings*. 2019; 12(1):27.
https://doi.org/10.3390/proceedings2019012027

**Chicago/Turabian Style**

Agresti, Iris, Gonzalo Carvacho, Davide Poderini, Leandro Aolita, Rafael Chaves, and Fabio Sciarrino.
2019. "Experimental Connection between the Instrumental and Bell Inequalities" *Proceedings* 12, no. 1: 27.
https://doi.org/10.3390/proceedings2019012027