Matrix product states and the quantum max-flow/min-cut conjectures
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 2d−2 between the quantum max-flow and the quantum min-cut.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Oct 01, 2018
- Source ID
- 10.1063/1.5026985
Entities
People
- Fulvio Gesmundo
- J. M. Landsberg
- Michael G Walter
Organizations
- Air Force Office of Scientific Research
- Dutch Research Council
- National Science Foundation
- Simons Foundation
- Stanford University
- Texas A&M University
- University of Copenhagen
- Villum Foundation