# IQI Weekly Seminar

**Abstract**: We consider the problem of certifying entanglement and nonlocality in

one-dimensional translation-invariant (TI) infinite systems when just

averaged near-neighbor correlators are available. Exploiting the triviality

of the marginal problem for 1D TI distributions, we arrive at a practical

characterization of the near-neighbor density matrices of multi-separable TI

quantum states. This allows us, e.g., to identify a family of separable

two-qubit states which only admit entangled TI extensions. For nonlocality

detection, we show that, when viewed as a vector in R^n, the set of boxes

admitting an infinite TI classical extension forms a polytope, i.e., a

convex set defined by a finite number of linear inequalities. Using DMRG, we

prove that some of these inequalities can be violated by distant parties

conducting identical measurements on an infinite TI quantum state. Both our

entanglement witnesses and our Bell inequalities can be used to certify

entanglement and nonlocality in large spin chains (namely, finite, and not

TI chains) via neutron scattering.

Our attempts at generalizing our results to TI systems in 2D and 3D lead us

to the virtually unexplored problem of characterizing the marginal

distributions of infinite TI systems in higher dimensions. In this regard,

we show that, for random variables which can only take a small number of

possible values (namely, bits and trits), the set of nearest (and

next-to-nearest) neighbor distributions admitting a 2D TI infinite extension

forms a polytope. This allows us to compute exactly the ground state energy

per site of any classical nearest-neighbor Ising-type TI Hamiltonian in the

infinite square or triangular lattice. Remarkably, some of these results

also hold in 3D.

In contrast, when the cardinality of the set of possible values grows (but

remaining finite), we show that the marginal nearest-neighbor distributions

of 2D TI systems are not described by a polytope or even a semi-algebraic

set. Moreover, the problem of computing the exact ground state energy per

site of arbitrary 2D TI Hamiltonians is undecidable.