Matrix product states and the quantum max-flow/min-cut conjectures
Quantum Physics
2018-10-19 v2 Algebraic Geometry
Abstract
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 2^{d-2} between the quantum max-flow and the quantum min-cut.
Cite
@article{arxiv.1801.09106,
title = {Matrix product states and the quantum max-flow/min-cut conjectures},
author = {Fulvio Gesmundo and J. M. Landsberg and Michael Walter},
journal= {arXiv preprint arXiv:1801.09106},
year = {2018}
}
Comments
12 pages, 3 figures - Final version accepted for publication on J. Math. Phys