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<p>We study and prove properties of the minimax formulation of the HRT holographic entanglement entropy formula, which involves finding the maximal-area surface on a timelike hypersurface, or time-sheet, and then minimizing over the choice of time-sheet. In this formulation, the homology condition is imposed at the level of the spacetime: the homology regions are spacetime volumes rather spatial regions. We show in particular that the smallest minimax homology region is the entanglement wedge. The minimax prescription suggests a way to construct a graph model for time-dependent states, a weighted graph on which min cuts compute HRT entropies. The existence of a graph model would imply that HRT entropies obey the same inequalities as RT entropies, in other words that the RT and HRT entropy cones coincide. Our construction of a graph model relies on the time-sheets obeying a certain “cooperating” property, which we show holds in some examples and for which we give a partial proof; however, we also find scenarios where it may fail. A video abstract is available at <a href="https://youtu.be/Ja-TPYNBujM">https://youtu.be/Ja-TPYNBujM</a>.</p>

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Minimax surfaces and the holographic entropy cone

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月13日

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  • Brianna Grado-White,
  • Guglielmo Grimaldi,
  • Matthew Headrick,
  • Veronika E. Hubeny

摘要

We study and prove properties of the minimax formulation of the HRT holographic entanglement entropy formula, which involves finding the maximal-area surface on a timelike hypersurface, or time-sheet, and then minimizing over the choice of time-sheet. In this formulation, the homology condition is imposed at the level of the spacetime: the homology regions are spacetime volumes rather spatial regions. We show in particular that the smallest minimax homology region is the entanglement wedge. The minimax prescription suggests a way to construct a graph model for time-dependent states, a weighted graph on which min cuts compute HRT entropies. The existence of a graph model would imply that HRT entropies obey the same inequalities as RT entropies, in other words that the RT and HRT entropy cones coincide. Our construction of a graph model relies on the time-sheets obeying a certain “cooperating” property, which we show holds in some examples and for which we give a partial proof; however, we also find scenarios where it may fail. A video abstract is available at https://youtu.be/Ja-TPYNBujM.