# Quantifying Tolerance of a Nonlocal Multi-Qudit State to Any Local Noise

## Abstract

**:**

## 1. Introduction

- to specify via parameters of an N-partite state the general analytical expressions for the noise tolerance of a nonlocal N-partite state (i) under ${S}_{1}\times \cdots \times {S}_{N}$-setting quantum correlation scenarios with any number of outcomes at each site and (ii) under all quantum correlation scenarios with arbitrary numbers of settings and outcomes per site;
- to derive new precise lower/upper bounds in d and N on the noise tolerances and the maximal amounts of tolerable local noise for: (i) an arbitrary nonlocal N-qudit state; (ii) the N-qudit Greenberger-Horne-Zeilinger (GHZ) state; (iii) the N-qubit W state and the N-qubit Dicke states and to analyse asymptotics of these precise new bounds for large N and $d.$

## 2. General $\mathbf{N}$-Partite Bell Inequalities

## 3. Quantum Violation

**Definition**

**1.**

- ${S}_{1}\times \cdots \times {S}_{N}$-setting nonlocal iff$${\mathrm{Y}}_{{S}_{1}\times \cdots \times {S}_{N}}^{\left(\rho \right)}>1$$$${\mathrm{Y}}_{{S}_{1}\times \cdots \times {S}_{N}}^{\left(\rho \right)}=1.$$
- (overall) nonlocal iff$${\mathrm{Y}}_{\rho}>1$$$${\mathrm{Y}}_{\rho}=1.$$

## 4. Tolerance to Any Local Noise

**Definition**

**2.**

**Definition**

**3.**

**Proposition**

**1.**

**Proof.**

## 5. General Bounds

## 6. N-Qudit GHZ State

#### N-Qubit Case

## 7. N-Qubit Dicke States

## 8. Conclusions

## Acknowledgments

## Conflicts of Interest

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Loubenets, E.R.
Quantifying Tolerance of a Nonlocal Multi-Qudit State to Any Local Noise. *Entropy* **2018**, *20*, 217.
https://doi.org/10.3390/e20040217

**AMA Style**

Loubenets ER.
Quantifying Tolerance of a Nonlocal Multi-Qudit State to Any Local Noise. *Entropy*. 2018; 20(4):217.
https://doi.org/10.3390/e20040217

**Chicago/Turabian Style**

Loubenets, Elena R.
2018. "Quantifying Tolerance of a Nonlocal Multi-Qudit State to Any Local Noise" *Entropy* 20, no. 4: 217.
https://doi.org/10.3390/e20040217