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Locality of edge states and entanglement spectrum from strong subadditivity

Kohtaro Kato and Fernando G. S. L. Brandão
Phys. Rev. B 99, 195124 – Published 14 May 2019
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  • INTRODUCTION
  • ASSUMPTION: UNIFORM AREA LAW
  • DEFINITION OF EDGE STATES AND THE MAIN…
  • ENTANGLEMENT SPECTRUM ON A CYLINDER
  • DISCUSSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We consider two-dimensional states of matter satisfying a uniform area law for entanglement. We show that the topological entanglement entropy is equal to the minimum relative entropy distance from the reduced state to the set of thermal states of local models. The argument is based on strong subadditivity of quantum entropy. For states with zero topological entanglement entropy, in particular, the formula gives locality of the states at the boundary of a region as thermal states of local Hamiltonians. It also implies that the entanglement spectrum of a two-dimensional region is equal to the spectrum of a one-dimensional local thermal state on the boundary of the region.

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    • Received 24 May 2018
    • Revised 17 April 2019

    DOI:https://doi.org/10.1103/PhysRevB.99.195124

    ©2019 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    AnyonsEntanglement entropyQuantum entanglementTopological phases of matter
    Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

    Authors & Affiliations

    Kohtaro Kato1,2 and Fernando G. S. L. Brandão2,3

    • 1Department of Physics, Graduate School of Science, University of Tokyo, Tokyo, Japan
    • 2Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
    • 3Google, Incorporated, 340 Main Street, Venice, California 90291, USA

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    Issue

    Vol. 99, Iss. 19 — 15 May 2019

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    Images

    • Figure 1
      Figure 1

      Region R, its boundary region X, and the complement R′. The size of each region Xi is specified by l.

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    • Figure 2
      Figure 2

      We consider a system on a 2D cylinder. We divide it into three regions Y, X, and Y′ so that X can be viewed as a 1D “boundary” of Y as in Fig. 1.

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    • Figure 3
      Figure 3

      Top: We choose X as the region around the physical boundary (the right edge). The entanglement spectrum on Y is the same as that of X. Bottom: In some cases, the reduced state on Y is almost independent of the length of the opposite side. Then the entanglement spectrum of Y is equivalent to the spectrum of X, which is an edge of another cylinder with shorter length.

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    • Figure 4
      Figure 4

      A schematic picture of reduction of the calculation of S(R)ρ. The topologically trivial ground state |ψ〉 can be created by a product state |0〉⊗N by applying a constant-depth local circuit. The time step goes from bottom to top and each box represents a unitary matrix Vi(ki) acting on subsystems represented by vertical lines. When we divide systems into R and Rc (by the dotted line), only boxes colored by black contribute to the entanglement. We can remove all gray boxes (URURc) without changing the entanglement entropy. Subsystems not acted on by black boxes are then uncorrelated to all other systems. The state on the remaining subsystems around the boundary is |ϕRRc〉.

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    • Figure 5
      Figure 5

      We can regard ∂R as a periodic spin ladder under coarse graining. |ϕRRc〉 is then represented as a MPS defined by two tensors A and C with a constant bond dimension. Each tensor has two legs corresponding to either spins in R or spins in Rc. By tracing out the outer indices, we obtain a MPO representation of the reduced state ϕR.

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