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.