Entanglement and Nonlocality in Infinite 1D Systems

Abstract

We consider the problem of detecting entanglement and nonlocality in one-dimensional (1D) infinite, translation-invariant (TI) systems when just near-neighbor information is available. This issue is deeper than one might think a priori, since, as we show, there exist instances of local separable states (classical boxes) which only admit entangled (non-classical) TI extensions. We provide a simple characterization of the set of local states of multi-separable TI spin chains and construct a family of linear witnesses which can detect entanglement in infinite TI states from the nearest-neighbor reduced density matrix. Similarly, we prove that the set of classical TI boxes forms a polytope and devise a general procedure to generate all Bell inequalities which characterize it. Using an algorithm based on matrix product states, we show how some of them can be violated by distant parties conducting identical measurements on an infinite TI quantum state. All our results can be easily adapted to detect entanglement and nonlocality in large (finite, not TI) 1D condensed matter systems.

Two-dimensional translation-invariant probability distributions: approximations, characterizations and no-go theorems

“…This conundrum is at the essence of the TI MARGINAL problem , where a number of probability distributions of finitely-many variables are provided and the task is to certify if they correspond to the marginals of a TI system. MARGINAL arises naturally at the intersection of quantum information science and condensed matter physics, when we try to determine whether the dynamical structure factors of a large spin system are compatible with an underlying Bell local quantum state [ 2 ].…”

“…Motivated by the desire of understanding the nature of quantum non-locality in 2D materials [ 2 ], in this paper we will considerably advance the above fundamental questions. We will show that, in scenarios where the random variables take a small number d of possible values, the MARGINAL problem for nearest-neighbour (for d = 2, 3) and next-to-nearest neighbour distributions (for d = 2) is exactly solvable.…”

Bell Correlations at Ising Quantum Critical Points

“…The spatial structure of entanglement, on the other hand, is rather revealed through the bipartite Schmidt decomposition, which captures entanglement at a many-body level. Developing further conceptual and technical tools to investigate non-local correlations in spatially structured many-body states is an important challenge for ongoing studies [24].…”

Bell correlations at finite temperature

“…For this reason, the implications of nonlocal correlations in systems consisting of a very large number of parties has been mostly lacking, despite remarkable results have been obtained throughout the years [27][28][29][30][31][32][33][34][35][36][37]. Interestingly, recent theoretical advances developed in the context of few-body Bell inequalities [38][39][40][41][42][43][44][45][46] have allowed for simple ways to detect these correlations in many-body systems. On the one hand, it has been shown that many-body observables such as the energy of the system could signal the presence of Bell correlations in the system [42].…”