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I ran into the terminology "entanglement barrier" perhaps in the context of MPS. such as this paper, https://arxiv.org/pdf/1904.11999.pdf, or others

I'd appreciate it if there is any explanation for this or a reference would be highly appreciated!

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Matrix product states can only efficiently represent states with low entanglement. To quantify entanglement in this context, it's useful to ask how the the entanglement entropy of a subsystem scales as one increases the size of that subsystem. Many states (such as ground states of gapped Hamiltonians) satisfy area-law scaling; the entanglement entropy scales with the size of the boundary of the subsystem. One can interpret area-law scaling as a sign that entanglement is local, and therefore only qubits near the boundary contribute to the entanglement entropy. Conversely, highly entangled states tend to exhibit volume-law scaling, where the entanglement entropy scales with the number of qubits in a subsystem, and therefore have long-range correlations. The entanglement barrier refers to the fact that MPS can only efficiently represent area-law entangled states.

For references, highly recommend Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks. This is a standard and accessible introduction, which also covers what you would need if you wanted to implement MPS techniques for numerical computations.

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