2012

Stability of the Area Law for the Entropy of Entanglement

Michalakis, Spyridon

Understand

Recent results on the stability of the spectral gap under general perturbations for frustration-free Hamiltonians, have motivated the following question: Does the entanglement entropy of quantum states that are connected to states satisfying an area law along gapped Hamiltonian paths, also satisfy an area law? We answer this question in the affirmative, combining recent advances in quasi-adiabatic evolution and Lieb-Robinson bounds with ideas from the proof of the 1D area law.

Built on

Hastings, M. B., Wen, X-G., “Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance”, Phys. Rev. B 72
2005
Earlier work this paper cites.
Hastings, M. B., Koma, T., “Spectral Gap and Exponential Decay of Correlations”, Commun. Math. Phys. 265
2006
Earlier work this paper cites.
Nachtergaele, B., Sims, R. “Lieb-Robinson Bounds and the Exponential Clustering Theorem”, Commun. Math. Phys. 265
2006
Earlier work this paper cites.
Hastings, M. B., “An Area Law for One Dimensional Quantum Systems”, J. Stat. Mechanics: Theory and Experiment, P08024 (2007)
2007
Earlier work this paper cites.
Osborne, T. J., “Simulating adiabatic evolution of gapped spin systems”, Phys. Rev. A, 75
2007
Earlier work this paper cites.

Similar

Schuch, N., Wolf, M. M., Verstraete, F., and Cirac, J. I., “Entropy Scaling and Simulability by Matrix Product States”, Phys. Rev. Lett. 100
2008
Cited alongside, same era.
Wolf, M. M., Verstraete, F., Hastings, M. B., and Cirac, J. I., “Area laws in quantum systems: mutual information and correlations”, Phys. Rev. Lett. 100
2008
Cited alongside, same era.
Hamza, E., Michalakis, S., Nachtergaele, B., and Sims, R., “Approximating the ground state of gapped quantum spin systems”, J. Math. Phys. 50
2009
Cited alongside, same era.
Bravyi, S., Hastings, M. B., Michalakis, S., “Topological quantum order: stability under local perturbations”, J. Math. Phys. 51
2010
Cited alongside, same era.
Eisert, J., Cramer, M, and Plenio, M. B., “Area laws for the entanglement entropy - a review”, Rev. Mod. Phys. 82
2010
Cited alongside, same era.
Arad, I., Landau, Z., and Vazirani, U., “An improved 1D area law for frustration-free systems”, arXiv:1111.2970v2 [quant-ph]
Original
Cited in the paper.
Brandao, F., Horodecki, M., “Exponential Decay of Correlations Implies Area Law”, arXiv:1206.2947v1 [quant-ph]
Original
Cited in the paper.
Hastings, M. B., “Quasi-adiabatic Continuation for Disordered Systems: Applications to Correlations, Lieb-Schultz-Mattis, and Hall Conductance”, arXiv:1001.5280v2 [math-ph]
Original
Cited in the paper.
Hastings, M. B., Michalakis, S., “Quantization of Hall Conductance For Interacting Electrons Without Averaging Assumptions”, arXiv:0911.4706v1 [math-ph]
Original
Cited in the paper.
Michalakis, S., “Entanglement and ground states of gapped Hamiltonians”, arXiv:0807.4741v1 [quant-ph]
Original
Cited in the paper.
Michalakis, S., Pytel, J., “Stability of Frustration-Free Hamiltonians”, arXiv:1109.1588v2 [quant-ph]
Original
Cited in the paper.

Then

Bravyi, S., Hastings, M. B., “A short proof of stability of topological order under local perturbations”, Commun. Math. Phys. 307
2011
Later among the works it cites.
Chen, X., Gu Z-C., Wen, X-G., “Complete classification of one-dimensional gapped quantum phases in interacting spin systems”, Phys. Rev. B 84
2011
Later among the works it cites.
Schuch, N., Perez-Garcia, D., and Cirac, J. I., “Classifying quantum phases using matrix product states and projected entangled pair states”, Phys. Rev. B. 84
2011
Later among the works it cites.
Bachmann, S., Michalakis, S., Nachtergaele, B., and Sims, R., “Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems ”, Commun. Math. Phys. 309
2012
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