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

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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.

OriginalsprogEngelsk
Artikelnummer102205
TidsskriftJournal of Mathematical Physics
Vol/bind59
Udgave nummer10
Sider (fra-til)1-11
ISSN0022-2488
DOI
StatusUdgivet - 1 okt. 2018

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