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

TL;DR

This paper investigates the geometry of matrix product states with periodic boundary conditions, providing examples where the quantum max-flow is strictly less than the quantum min-cut, revealing fundamental inequalities in quantum network models.

Contribution

It presents three infinite sequences of examples demonstrating the quantum max-flow/min-cut inequality failure in specific graph configurations and bond dimensions.

Findings

Quantum max-flow is less than quantum min-cut for certain graph structures.

Inequality occurs for infinitely many bond dimensions in a 4-cycle graph.

A gap of at least 2^{d-2} exists between max-flow and min-cut for 2d-cycle graphs with bond dimension two.

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.