Fetching the paper…

Reading the bibliography…

2013

An area law and sub-exponential algorithm for 1D systems

Arad, Itai, Kitaev, Alexei, Landau, Zeph et al.

Original
PDF
Post on X

Understand

We give a new proof for the area law for general 1D gapped systems, which exponentially improves Hastings' famous result \cite{ref:Has07}.

Specifically, we show that for a chain of d-dimensional spins, governed by a 1D local Hamiltonian with a spectral gap \eps>0, the entanglement entropy of the ground state with respect to any cut in the chain is upper bounded by $O{\frac{\log^3 d}{\eps}}$.
Our approach uses the framework Arad et al to construct a Chebyshev-based AGSP (Approximate Ground Space Projection) with favorable factors.
However, our construction uses the Hamiltonian directly, instead of using the Detectability lemma, which allows us to work with general (frustrated) Hamiltonians, as well as slightly improving the $1/\eps$ dependence of the bound in Arad et al.

An area law and sub-exponential algorithm for 1D systems · Around

Built on

M. B. Hastings, “An area law for one-dimensional quantum systems,” Journal of Statistical Mechanics: Theory and Experiment
2007
Earlier work this paper cites.
D. Aharonov, I. Arad, Z. Landau, and U. Vazirani, “The detectability lemma and quantum gap amplification,” in STOC ’09: Proceedings of the 41st annual ACM symposium on Theory of computing, arXiv:0811.3412
Original
2009
Earlier work this paper cites.

Similar

J. Eisert, M. Cramer, and M. B. Plenio, “Colloquium: Area laws for the entanglement entropy,” Rev. Mod. Phys
2010
Cited alongside, same era.
D. Aharonov, I. Arad, Z. Landau, and U. Vazirani, “The 1d area law and the complexity of quantum states: A combinatorial approach,” in 2011 IEEE 52st Annual Symposium on Foundations of Computer Science
2011
Cited alongside, same era.

Then

D. Aharonov, I. Arad, U. Vazirani, and Z. Landau, “The detectability lemma and its applications to quantum hamiltonian complexity,” New Journal of Physics
2011
Later among the works it cites.
I. Arad, Z. Landau, and U. Vazirani, “Improved one-dimensional area law for frustration-free systems,” Phys. Rev. B
2012
Later among the works it cites.

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…