Network Nonlocality via Rigidity of Token-Counting and Color-Matching

arXiv:2011.02769
Some entangled quantum state correlations cannot be reproduced classically. Such correlations are called Nonlocal. This notion extendes to Network Nonlocality by considering several entangled states shared in a network. Here, we introduce two families of strategies to produce network nonlocal correlations. In the first one, called Token-Counting (TC), each source distributes a fixed number of tokens and each party counts the received tokens number. In the second one, called Color-Matching (CM), each source takes a color and a party checks if the color of neighboring sources match. Using graph theoretic tools and Finner's inequality, we show that TC and CM distributions are rigid in many networks: there is essentially a unique classical strategy to simulate such correlations. Using this rigidity property, we show that certain quantum TC and CM strategies produce correlations that cannot be produced classicality. This leads us to several examples of Network Nonlocality without input. These examples involve creation of coherence throughout the whole network, a fingerprint of genuine forms of Network Nonlocality.