Matrix product states and the quantum max-flow/min-cut conjectures Academic Article uri icon

abstract

  • © 2018 Author(s). In this note, we discuss the geometry of matrix product states with periodic boundary conditions and provide three infinite sequences of examples where the quantum max-flow is strictly less than the quantum min-cut. In the first, we fix the underlying graph to be a 4-cycle and verify a prediction of Hastings that inequality occurs for infinitely many bond dimensions. In the second, we generalize this result to a 2d-cycle. In the third, we show that the 2d-cycle with periodic boundary conditions gives inequality for all d when all bond dimensions equal two, namely, a gap of at least 2d−2 between the quantum max-flow and the quantum min-cut.

altmetric score

  • 0.75

author list (cited authors)

  • Gesmundo, F., Landsberg, J. M., & Walter, M.

citation count

  • 2

publication date

  • October 2018