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

Abstract

We study the properties of the set of marginal distributions of infinite translation-invariant systems in the two-dimensional square lattice. In cases where the local variables can only take a small number d of possible values, we completely solve the marginal or membership problem for nearest-neighbours distributions (d = 2, 3) and nearest and next-to-nearest neighbours distributions (d = 2). Remarkably, all these sets form convex polytopes in probability space. This allows us to devise an algorithm to compute the minimum energy per site of any TI Hamiltonian in these scenarios exactly. We also devise a simple algorithm to approximate the minimum energy per site up to arbitrary accuracy for the cases not covered above. For variables of a higher (but finite) dimensionality, we prove two no-go results. To begin, the exact computation of the energy per site of arbitrary TI Hamiltonians with only nearest-neighbour interactions is an undecidable problem. In addition, in scenarios with d≥2947, the boundary of the set of nearest-neighbour marginal distributions contains both flat and smoothly curved surfaces and the set itself is not semi-algebraic. This implies, in particular, that it cannot be characterized via semidefinite programming, even if we allow the input of the programme to include polynomials of nearest-neighbour probabilities.

Entanglement marginal problems

“…sym ) ⊗2k , ρ(L) 0, tr {(1,1),(2,1)} L (ρ (L) ) = tr L {(1,k),(2,k)} (ρ (L) ), SWAPρ (L) SWAP † = ρ(L) , (26) where SWAP is the operator permuting the Hilbert spaces of sites (0, y) and (1, y), for y = 0, ..., k − 1. The convergence of this hierarchy fol-lows from the proof of proposition 4 and the result, proven in [38], that any distribution P Î (a Î ) satisfying reflection symmetry along the vertical axis in addition to LTI admits a TI extension that is also symmetric with respect to vertical reflections. We arrive at the following theorem.…”

“…The result in [38] on the classical marginal problem generalizes to square lattices of all spatial dimensions. We arrive at the following result.…”

“…In [38], it is shown that any reflection-invariant distribution pÎ (φ Î ) defined on a 2 × 2 plaquette admits an extension p(φ) to the whole square lattice that is also TI and RI. It follows that ρ admits a TI, RI separable given by p(φ).…”

“…The characterization of TI distributions with reflection symmetry presented in [38] can be easily extended to arbitrary spatial dimensions. The general result would state that any probability distribution P for the variables of the sites {(x 1 , x 1 , ..., x D ) : x 1 ∈ {1, ..., k}, x 2 , ..., x D ∈ {1, 2}} for a hypercubic lattice admits a TI extension with symmetry under the reflection of the last D−1 axes iff P is LTI along the first axis and symmetric under the inversion of the rest.…”

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].…”