# Entanglement 25 Years after Quantum Teleportation: Testing Joint Measurements in Quantum Networks

## Abstract

**:**

## 1. Introduction

## 2. Quantum Teleportation and High-Impact Journals

## 3. The Bell-State Measurement in Quantum Networks

## 4. The Elegant Joint Measurement on Two Qubits

## 5. Quantum Correlation from Singlets and the EJM in the Triangle Configuration

## 6. Is ${\mathit{p}}_{\mathit{tr}}(\mathit{a},\mathit{b},\mathit{c})$ Three-Local?

#### A Natural but Asymmetric Three-Local Model

## 7. Consequences of a Non-Three-Local Quantum Triangle

## 8. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References and Notes

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**Figure 1.**Example of a quantum network. Each edge represents a resource shared by the connected nodes. The resource are entangled quantum states, or, in order to compare with classical networks, correlated local variables (i.e., shared randomness). In this paper we consider only cases where inputs are provided to parties connected by a single edge.

**Figure 2.**(N-1)-local scenario in a line [23]. The ${\lambda}_{j}$’s represent independent quantum states, or, in the classical scenario used for comparison, random independent local variables. Only the first and last parties get inputs, x and y respectively.

**Figure 3.**The triangle configuration for three parties [23]. Each pair of parties shares either a quantum state and performs quantum measurements—quantum scenario, or shares independent random variables $\alpha $, $\beta $ and $\gamma $ and outputs a function of the random variables to which they have access. Notice that the three random variables are only used locally, hence the terminology three-local scenario. The “quantum grail” is to find a quantum scenario (without external inputs) leading to a probability $p(a,b,c)$ which can’t be reproduced by any three-local scenario.

**Table 1.**The eight lines correspond to the eight possible combinations of values of ${\alpha}_{2}$, ${\beta}_{2}$ and ${\gamma}_{2}$ (first three columns). The next three columns indicate Alice, Bob and Charlie’s outputs. The seventh column indicates the probability of the corresponding line and the last two columns the probability that $a=b$ and $a=b=c$, respectively.

${\mathit{\alpha}}_{2}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{\gamma}}_{2}$ | a | b | c | P | Prob ($\mathit{a}=\mathit{b}$) | Prob ($\mathit{a}=\mathit{b}=\mathit{c}$) |
---|---|---|---|---|---|---|---|---|

0 | 0 | 0 | ${\beta}_{1}|{\gamma}_{1}$ | ${\alpha}_{1}|{\gamma}_{1}$ | ${\alpha}_{1}|{\beta}_{1}$ | ${\overline{q}}^{3}$ | 7/16 | 13/64 |

0 | 0 | 1 | ${\gamma}_{1}$ | ${\gamma}_{1}$ | ${\alpha}_{1}|{\beta}_{1}$ | ${\overline{q}}^{2}q$ | 1 | 1/4 |

0 | 1 | 0 | ${\beta}_{1}$ | ${\alpha}_{1}|{\gamma}_{1}$ | ${\beta}_{1}$ | ${\overline{q}}^{2}q$ | 1/4 | 1/4 |

0 | 1 | 1 | ${\beta}_{1}|{\gamma}_{1}$ | ${\gamma}_{1}$ | ${\beta}_{1}$ | $\overline{q}{q}^{2}$ | 5/8 | 1/4 |

1 | 0 | 0 | ${\beta}_{1}|{\gamma}_{1}$ | ${\alpha}_{1}$ | ${\alpha}_{1}$ | ${\overline{q}}^{2}q$ | 1/4 | 1/4 |

1 | 0 | 1 | ${\gamma}_{1}$ | ${\alpha}_{1}|{\gamma}_{1}$ | ${\alpha}_{1}$ | $\overline{q}{q}^{2}$ | 5/8 | 1/4 |

1 | 1 | 0 | ${\beta}_{1}$ | ${\alpha}_{1}$ | ${\alpha}_{1}|{\beta}_{1}$ | $\overline{q}{q}^{2}$ | 1/4 | 1/4 |

1 | 1 | 1 | ${\beta}_{1}|{\gamma}_{1}$ | ${\alpha}_{1}|{\gamma}_{1}$ | ${\alpha}_{1}|{\beta}_{1}$ | ${q}^{3}$ | 7/16 | 13/64 |

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Gisin, N.
Entanglement 25 Years after Quantum Teleportation: Testing Joint Measurements in Quantum Networks. *Entropy* **2019**, *21*, 325.
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**AMA Style**

Gisin N.
Entanglement 25 Years after Quantum Teleportation: Testing Joint Measurements in Quantum Networks. *Entropy*. 2019; 21(3):325.
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**Chicago/Turabian Style**

Gisin, Nicolas.
2019. "Entanglement 25 Years after Quantum Teleportation: Testing Joint Measurements in Quantum Networks" *Entropy* 21, no. 3: 325.
https://doi.org/10.3390/e21030325