TY - JOUR

T1 - Matrix product states and the quantum max-flow/min-cut conjectures

AU - Gesmundo, Fulvio

AU - Landsberg, J. M.

AU - Walter, Michael

PY - 2018/10/1

Y1 - 2018/10/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85055181203&partnerID=8YFLogxK

U2 - 10.1063/1.5026985

DO - 10.1063/1.5026985

M3 - Journal article

AN - SCOPUS:85055181203

VL - 59

SP - 1

EP - 11

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 10

M1 - 102205

ER -