Many-body scenarios offer an essentially virgin landscape to test the limits of quantum theory. The lack of control in manipulating these systems, together with the astronomical numbers of sites involved, make state tomography an impossible task. Hence the assumption of an underlying quantum state for the whole system is more a matter of faith than a testable physical prediction.

Tensor network states

To cope with the difficulty of describing general many-body quantum states, solid state physicists postulate that, in the low temperature regime, condensed matter states are accurately described by a sub-class of quantum states dubbed Tensor Network States (TNS). The complexity of such states, measured by a parameter known as the bond dimension, is conjectured to be relatively low in natural macroscopic systems.

Taking the above notion at face value, the study of the correlations achievable with general quantum states becomes irrelevant, because just a subset of them are physically realizable. It is hence crucial to understand how TNS differ from general (unphysical) quantum states. Given that many condensed matter systems can only be probed via neutron scattering, one wonders how the properties of TNS would manifest in such experiments.

In a past pre-print [1], we proved that Matrix Product States (MPS), the class of TNS used to model 1-dimensional spin chains, satisfy non-trivial structural properties. More specifically, we found that there exist local operators which annihilate all of them, and local operators which decouple (cut) the particles where they act from the spin chain while at the same time join (glue) the two loose ends back again into an MPS, see the Figure.

Similarly, we presented families of local observables whose expectation values, accessible via neutron scattering, are non-negative when evaluated by convex combinations of MPS of a given bond dimension. A negative expectation value of such bond dimension witnesses would signify that the underlying state is actually more complex than originally envisaged.

Entanglement and nonlocality in infinite spin chains

A usual simplification in the study of condensed mater systems, which becomes exact in quantum field theory, is that they are composed by infinitely many sites and that the thermal state of the system is invariant under spatial translations. This symmetry, although weak, enforces a pattern of regularity which sometimes allows one to infer global properties of the whole chain from local observations.

In this spirit, we studied how to characterize entanglement and nonlocality in 1-D translation-invariant infinite spin-chains from local measurements [2]. We found instances of TI quantum states with a nearest-neighbor state that, despite being separable, allows one to conclude that the whole chain is entangled. We also solved the problem of determining whether a translation-invariant box for infinitely many parties is nonlocal (see Black Box Quantum Theory), given experimental data gathered by finitely many parties in a bounded region of the chain. In both cases, the degree of violation of an entanglement witness or a Bell inequality is inversely proportional to the block size of the chain which we need to access in order to exploit entanglement or nonlocality resources.



Our attempts to extend our results to higher spatial dimensions led us to tackle the classical marginal problem in 2D [3]. Consider an infinite square lattice, and suppose that each site gives one access to a unique random variable. A finite number of agents exploring this lattice would only have access to the random variables in their immediate surroundings. If this lattice happened to be translation-invariant, then the probability distributions of the variables probed by the agents would be constrained by this symmetry in a non-trivial way. The marginal problem consists in identifying all constraints which such marginal distributions are subject to, or, more poetically, to paint what these agents would see.


For instance, suppose that agents 1 and 2 in the Figure above estimate the probability distributions of the variables in one site and the site above, Pv(a,b), and one site and the site on its right, Ph(a,b). These distributions are so important in statistical physics that they have a name: they are called nearest-neighbor distributions. Given Pv, Ph the marginal problem for nearest-neighbor distributions consists in determining whether there exists a 2D translation-invariant system with Pv, Phas marginals. Similarly, we could have considered whether the 4-site distribution estimated by agent 3 is compatible with translation-invariance.

The marginal problem is solvable for 1D systems: given any distribution of a (finite) number of sites in a chain, there exists an algorithm with a fixed number of steps to decide if such a distribution could have arisen from a 1D translation-invariant system. It was by exploiting the solution of the 1D marginal problem that we managed to characterize nonlocality and entanglement in 1D quantum systems [2]. In two spatial dimensions or more, the marginal problem was essentially unexplored. Hence we attacked it.

In [3] we find that the marginal problem in 2D can be solved for nearest-neighbor distributions when the random variables can just take d=3possible values. It can also be solved for nearest-neighbor and next-to-nearest-neighbor distributions when d=2. In such scenarios, it is possible to compute the minimum energy per site of arbitrary translation-invariant Hamiltonians analytically. In the latter, we can even calculate the number of lattice configurations achieving the minimum energy-per site.

When viewed in probability space, the set of 2D TI marginal distributions in all solvable scenarios (including 1D) is a polytope, i.e., a set of points described by a finite number of linear inequalities, see Figure above, left. When d is of the order of thousands, the boundary of the set of neatest-neighbor marginals contains both flat and smoothly curved pieces, see the Figure above, right. Actually, for d=2947, this set cannot even be described by a finite number of polynomial inequalities (namely, it is not semi-algebraic) [3]. To make matters worse, for arbitrary arbitrary d, the minimum energy per site of a nearest-neighbor translation-invariant Hamiltonian is an incomputable function. That is, no Turing machine can evaluate it in all instances.

All the above makes exploring the entanglement and non-locality of 2D systems a challenging task. But we won’t give up!

[1] M. Navascués and T. Vértesi, Bond dimension witnesses and the structure of homogeneous matrix product states, Quantum 2, 50 (2018).

[2] Z. Wang, S. Singh, M. Navascués, Entanglement and Nonlocality of 1D Macroscopic Systems, Phys. Rev. Lett. 118, 230401 (2017).    

[3] Z. Wang and M. Navascués, Two-dimensional translation-invariant probability distributions: approximations, characterizations and no-go theorems, Proc. R. Soc. A-Math. Phys. Eng. Sci. 474, 20170822 (2018), arXiv:1703.05640.

Additional readings

D. Pérez-García, F. Verstraete, M.M. Wolf, and J.I. Cirac, Matrix product state representations, Quantum Inf. Comput., 7:401 (2007).