# Non-Classical Correlations in n-Cycle Setting

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## Abstract

**:**

## 1. Introduction

- The KS boxes were motivated by KCBS non-contextuality inequality [18].
- The KCBS non-contextuality inequality belongs to the same family of inequalities as the CHSH inequality [9].
- The maximum value of CHSH inequality for no-signalling theories is provided by PR boxes [16].
- The PR box can be simulated by KS box for $p=\frac{1}{2}$ [18].

#### Paper Structure

## 2. Simulating KS Box

**Definition**

**1.**

- 1.
- $a=b$ if $x=y$, and
- 2.
- $a.b=0$ if $x\ne y.$

**Proposition**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

## 3. Analysing n-Cycle Non-Contextuality Inequalities

#### 3.1. KCBS Inequality

- ${P}_{i}$ and ${P}_{i}+1$ are compatible and
- ${P}_{i}$ and ${P}_{i+1}$ are exclusive.

#### 3.2. CHSH Inequality

#### 3.3. Analysing the Generalised KCBS Inequality

**Proposition**

**2.**

**Proof.**

**Remark**

**1.**

#### 3.4. Analysing Chained Bell Inequalities

**Proposition**

**3.**

**Proof.**

**Remark**

**2.**

## 4. Simulating PR Box

**Definition**

**2.**

**Proposition**

**4.**

**Proof.**

**Remark**

**3.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proposition**

**A1.**

**Proof.**

## References

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**Figure 1.**KS box is a bipartite no-signalling box. The value of a does not depend on y and similarly b does not depend on x. The box exhibits nonlocal correlations.

**Figure 2.**Chart ${C}_{2}$ for a five-dimensional KS box corresponds to two “1s” and three “0s”. The red entries correspond to inputs and the outputs are in green. The above chart fails to simulate the KS box statistics when the inputs are 2 and 5.

**Figure 3.**The simulation efficiency has been plotted here as a function of the dimension of the KS box for various marginal probabilities, p. It can be seen that the simulation efficiency decreases with dimension for a particular p.

**Figure 4.**The exclusivity graph corresponding to the KCBS inequality is a pentagon. The inequality involves five events of type $\left(1|i\right)$ where $i\in \{1,2,3,4,5\}.$ The bound on the inequality for non-contextual hidden variable theories is $2.$ Quantum theory achieves up to $\sqrt{5}$ and thus manifests the contextual nature of quantum theory.

**Figure 5.**The condition for the quantum violation of the odd n-cycle generalisation of KCBS inequality is computed. Lower bound on ${\rho}_{33}$ for odd n-cycle graph has been plotted as a function of n. The set of states which can violate the KCBS inequality corresponding to optimal measurement setting shrinks as we increase n.

**Figure 6.**Here, we plot the lower bound on the difference of extremal eigenvalues of a two qubit density matrix as a function of even values of n. The set of two qubit quantum states, which could potentially violate chained Bell inequality (as our is necessary and not sufficient), shrinks as we increase $n.$ In the infinite n scenario, the only two qubit state that might violate the inequality is Bell state!

**Figure 7.**We look at KS box probabilities in various regimes. Note that the quantum probability approaches the PR box limit faster than classical probability as we increase the number of cycles.

Dimension | Marginal Probability | Simulation Efficiency |
---|---|---|

5 | 0.4 | 0.92 |

7 | 0.4 | 0.893878 |

9 | 0.4 | 0.881481 |

11 | 0.4 | 0.87438 |

13 | 0.4 | 0.869822 |

15 | 0.4 | 0.866667 |

17 | 0.4 | 0.862976 |

**Table 2.**The table displays the joint probabilities for an n-dimensional KS${}_{p}$ box. Note that each of the blocks along the diagonal are same and similarly all the off diagonal blocks are same. Within a block, the top left element is the probability of getting $(0,0)$, top right signifies the probability of getting $(0,1)$, bottom left indicates the corresponding value for $(1,0)$ and, the probability for $(1,1)$ is indicated by the bottom right entry.

x | 1 | 2 | ⋯ | n | ||||||
---|---|---|---|---|---|---|---|---|---|---|

y | ||||||||||

1 | $1-p$ | 0 | $1-2p$ | p | ⋱ | $1-2p$ | p | |||

0 | p | p | 0 | p | 0 | |||||

2 | $1-2p$ | p | $1-p$ | 0 | ⋱ | $1-2p$ | p | |||

p | 0 | 0 | p | p | 0 | |||||

⋮ | ⋱ | ⋱ | ⋱ | ⋱ | ||||||

n | $1-2p$ | p | $1-2p$ | p | ⋱ | $1-p$ | 0 | |||

p | 0 | p | 0 | 0 | p |

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Bharti, K.; Ray, M.; Kwek, L.-C.
Non-Classical Correlations in *n*-Cycle Setting. *Entropy* **2019**, *21*, 134.
https://doi.org/10.3390/e21020134

**AMA Style**

Bharti K, Ray M, Kwek L-C.
Non-Classical Correlations in *n*-Cycle Setting. *Entropy*. 2019; 21(2):134.
https://doi.org/10.3390/e21020134

**Chicago/Turabian Style**

Bharti, Kishor, Maharshi Ray, and Leong-Chuan Kwek.
2019. "Non-Classical Correlations in *n*-Cycle Setting" *Entropy* 21, no. 2: 134.
https://doi.org/10.3390/e21020134