Entanglement and nonlocality in infinite quantum systems
Scientists probing the sites of an infinite quantum system
Imagine a number of scientists participating in a space exploratory mission. Our heroes board separate vessels and depart from Earth in different directions. Each space ship is equipped with lab equipment, so the astronauts can conduct experiments during the course of their journey. We do not need to specify the exact nature of such experiments, but, to fix ideas, one could picture the explorers performing local measurements on the vacuum of a quantum field theory.
The astronauts find that the results of the experiments in the different vessels are statistically correlated. Furthermore, they do not seem to depend on the precise positions of the rockets, but just on the relative positions between them. These lab experiences lead them to postulate that this property, translation invariance, must hold everywhere, and not just within the region of space that they have been exploring so far. To model physically this assumption, we would posit that the experiments are tapping the sites of an infinite translation invariant system.
In a recent paper, IQOQI members Z. Wang, S. Singh and M. Navascués prove that, in such a predicament, the space travelers could conclude that the overall state of the infinite system which they are probing is entangled or nonlocal, even when they were not directly observing entanglement or nonlocality.
This is so because, as the authors show, there exist local quantum separable states with the property that any translation invariant extension of them must be necessarily entangled. Similarly, there exist multipartite correlations achievable within translation-invariant quantum systems which, although simulatable via classical devices, are not compatible with the behavior exhibited by any classical infinite translation-invariant object.
In their work, Wang et al. derive a family of entanglement witnesses to detect entanglement in one-dimensional (1D) infinite translation-invariant systems. They also prove that the local statistics arising from infinite 1D translation-invariant classical systems live inside a polyhedron in probability space, for which they provide an analytic characterization. This is a surprising discovery, since, in 2D, the corresponding set of probability vectors has both flat and smoothly curved boundaries and the laws of logic are not powerful enough to characterize it completely.
To obtain their results, the researchers relied on a number of ideas from condensed matter theory. It is intriguing where this unusual connection between the foundations of quantum mechanics and solid state physics will lead them in the future.