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

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.Comment: 12 pages, 3 figures - Final version accepted for publication on J. Math. Phy

Quantum bit threads

We give a bit thread prescription that is equivalent to the quantum extremal surface prescription for holographic entanglement entropy. Our proposal is inspired by considerations of bit threads in doubly holographic models, and equivalence is established by proving a generalisation of the Riemannian max-flow min-cut theorem. We explore our proposal's properties and discuss ways in which islands and spacetime are emergent phenomena from the quantum bit thread perspective.Comment: 24 pages, 5 figures; v2: minor improvements made and new appendix added; v3: version for publication with no significant change

Perfect Quantum Network Communication Protocol Based on Classical Network Coding

This paper considers a problem of quantum communication between parties that are connected through a network of quantum channels. The model in this paper assumes that there is no prior entanglement shared among any of the parties, but that classical communication is free. The task is to perfectly transfer an unknown quantum state from a source subsystem to a target subsystem, where both source and target are formed by ordered sets of some of the nodes. It is proved that a lower bound of the rate at which this quantum communication task is possible is given by the classical min-cut max-flow theorem of network coding, where the capacities in question are the quantum capacities of the edges of the network.Comment: LaTeX2e, 10 pages, 2 figure

Quantum Capacities for Entanglement Networks

We discuss quantum capacities for two types of entanglement networks: $\mathcal{Q}$ for the quantum repeater network with free classical communication, and $\mathcal{R}$ for the tensor network as the rank of the linear operation represented by the tensor network. We find that $\mathcal{Q}$ always equals $\mathcal{R}$ in the regularized case for the samenetwork graph. However, the relationships between the corresponding one-shot capacities $\mathcal{Q}_1$ and $\mathcal{R}_1$ are more complicated, and the min-cut upper bound is in general not achievable. We show that the tensor network can be viewed as a stochastic protocol with the quantum repeater network, such that $\mathcal{R}_1$ is a natural upper bound of $\mathcal{Q}_1$. We analyze the possible gap between $\mathcal{R}_1$ and $\mathcal{Q}_1$ for certain networks, and compare them with the one-shot classical capacity of the corresponding classical network