Citation: Ge, Yimin, and Jens Eisert. "Area laws and efficient descriptions of quantum many-body states." New Journal of Physics 18.8 (2016): 083026.
Web: https://arxiv.org/abs/1411.2995
Tags: Lieb-Robinson, Information-theory, Tensor-networks
This paper proves that the vast majority of states with area laws in D>1 dimensions do not admit efficient descriptions. In particular, they cannot be prepared efficiently on a quantum computer, they are not the ground states of local Hamiltonians, and they cannot be described in terms of a tensor network like PEPS or MERA.
This is in very sharp contrast to the case of D=1. In this case, the classification of gapped quantum systems went as follows. First, Hastings proved his area law for 1D quantum systems:
> Hastings, Matthew B. "An area law for one-dimensional quantum systems." Journal of statistical mechanics: theory and experiment 2007.08 (2007): P08024.
It was then well-known that every 1D quantum state with an area can be described as an MPS. Using this efficient rigid description, it is then possible to get a complete claudication of gapped 1D quantum systems:
> Chen, Xie, Zheng-Cheng Gu, and Xiao-Gang Wen. "Classification of gapped symmetric phases in one-dimensional spin systems." Physical Review B—Condensed Matter and Materials Physics 83.3 (2011): 035107.
> Schuch, Norbert, David Pérez-García, and Ignacio Cirac. "Classifying quantum phases using matrix product states and projected entangled pair states." Physical Review B—Condensed Matter and Materials Physics 84.16 (2011): 165139.
> Szehr, Oleg, and Michael M. Wolf. "Connected components of irreducible maps and 1D quantum phases." Journal of Mathematical Physics 57.8 (2016).
The hope was that a similar procedure could be followed in D>1 dimensions. This paper highlights the fact that there are more to physical states than just area laws. There is some other property which characterizes those states which admit efficient description.