Canonical partial ordering from min-cuts and quantum entanglement in random tensor networks

• URL: http://arxiv.org/abs/2506.23894v1
• Date: Mon, 30 Jun 2025 14:26:40 GMT
• Title: Canonical partial ordering from min-cuts and quantum entanglement in random tensor networks
• Authors: Miao Hu, Ion Nechita,
• Abstract summary: We show that the emphfinite correction to entanglement R'enyi entropy arising from the degeneracy of the min-cuts is given by the number of emphorder morphisms<n>We also show that the number of order morphisms corresponds to moments of a graph-dependent measure which generalizes the free Bessel law in some special cases in free probability theory.
• Score: 5.882601249202242
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The \emph{max-flow min-cut theorem} has been recently used in the theory of random tensor networks in quantum information theory, where it is helpful for computing the behavior of important physical quantities, such as the entanglement entropy. In this paper, we extend the max-flow min-cut theorem to a relation among different \emph{partial orders} on the set of vertices of a network and introduce a new partial order for the vertices based on the \emph{min-cut structure} of the network. We apply the extended max-flow min-cut theorem to random tensor networks and find that the \emph{finite correction} to the entanglement R\'enyi entropy arising from the degeneracy of the min-cuts is given by the number of \emph{order morphisms} from the min-cut partial order to the partial order induced by non-crossing partitions on the symmetric group. Moreover, we show that the number of order morphisms corresponds to moments of a graph-dependent measure which generalizes the free Bessel law in some special cases in free probability theory.

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