Abstract
The Entanglement contour function quantifies the contribution from each degree of freedom in a region mathcal{A} to the entanglement entropy {S}_{mathcal{A}} . Recently in [1] the author gave two proposals for the entanglement contour in two-dimensional theories. The first proposal is a fine structure analysis of the entanglement wedge, which applies to holographic theories. The second proposal is a claim that for general two-dimensional theories the partial entanglement entropy is given by a linear combination of entanglement entropies of relevant subsets inside mathcal{A} . In this paper, we further study the partial entanglement entropy proposal by showing that it satisfies all the rational requirements proposed previously. We also extend the fine structure analysis from vacuum AdS space to BTZ black holes. Furthermore, we give a simple prescription to generate the local modular flows for two-dimensional theories from only the entanglement entropies without refer to the explicit Rindler transformations.
Highlights
Without loss of generality, we only consider α to be connected
Since we only used the construction of the proposal (2.1) and the general inequalities satisfied by entanglement entropies, the partial entanglement entropy (PEE) proposal should satisfy the above 6 properties in general two-dimensional theories
Since the entanglement contour function respect the symmetries, for highly symmetric configurations2 the contour functions may only depend on one coordinate
Summary
We focus on the entanglement contour functions for single intervals in 2dimensional theories. When the subset A2 shares one boundary with A, it is easy to see that the sA(A2) calculated by the proposal is just half of the mutual entanglement entropy between A2 and Ac, sA(A2). Using SA = SAc and SA1∪A3 = SA2∪Ac , we can write the lower bound (2.12) as This means the sA(A2) calculated by the proposal is always larger than half of the mutual information between A2 and Ac. Though the PEE proposal (2.1) satisfies all the physical requirements proposed in [16], it is not justified because those requirements are not enough to uniquely determine the PEE. Assuming that the PEE is a linear combination of all the subset entanglement entropies, i.e. Where the coefficients are constants that are independent from the choice of the partition. Imposing the requirements of additivity (2.6) and normalization, one can uniquely determine the coefficients in the ansatz and get (2.1)
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