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Nonviolation of Bell's Inequality in Translation Invariant Systems

The nature of quantum correlations in strongly correlated systems has been a subject of intense research. In particular, it has been realized that entanglement and quantum discord are present at quantum phase transitions and able to characterize it. Surprisingly, it has been shown for a number of different systems that qubit pairwise states, even when highly entangled, do not violate Bell's inequalities, being in this sense local. Here we show that such a local character of quantum correlations is in fact general for translation invariant systems and has its origins in the monogamy trade-off obeyed by tripartite Bell correlations. We illustrate this result in a quantum spin chain with a soft breaking of translation symmetry. In addition, we extend the monogamy inequality to the $N$-qubit scenario, showing that the bound increases with $N$ and providing examples of its saturation through uniformly generated random pure states.