Genuine multientropy and holography

Is bipartite entanglement sufficient for holography? Through the analysis of the Markov gap, it is known that the answer is no. In this paper, we give a new perspective on this issue from a different angle using a multientropy. We define a genuine <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi mathvariant="monospace">q</a:mi></a:math>-partite multientropy from a <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi mathvariant="monospace">q</d:mi></d:math>-partite multientropy by subtracting appropriate linear combinations of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mrow><g:mover accent="true"><g:mrow><g:mi mathvariant="monospace">q</g:mi></g:mrow><g:mrow><g:mo stretchy="false">˜</g:mo></g:mrow></g:mover></g:mrow></g:math>-partite multientropies for <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mrow><l:mrow><l:mover accent="true"><l:mrow><l:mi mathvariant="monospace">q</l:mi></l:mrow><l:mrow><l:mo stretchy="false">˜</l:mo></l:mrow></l:mover></l:mrow><l:mo><</l:mo><l:mi mathvariant="monospace">q</l:mi></l:mrow></l:math>, in such a way that the genuine <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mi mathvariant="monospace">q</r:mi></r:math>-partite multientropy vanishes for all <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mrow><u:mover accent="true"><u:mrow><u:mi mathvariant="monospace">q</u:mi></u:mrow><u:mrow><u:mo stretchy="false">˜</u:mo></u:mrow></u:mover></u:mrow></u:math>-partite entangled states. After studying several aspects, we apply it to black holes and holography. For the application to black holes, we see that such a genuine <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:mi mathvariant="monospace">q</z:mi></z:math>-partite multientropy is important only after the Page time. For the application to holography, we prove that nonbipartite multientropies are always positive and <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mi mathvariant="script">O</cb:mi><cb:mrow><cb:mo stretchy="false">(</cb:mo><cb:mrow><cb:mn>1</cb:mn><cb:mo>/</cb:mo><cb:msub><cb:mi>G</cb:mi><cb:mi>N</cb:mi></cb:msub></cb:mrow><cb:mo stretchy="false">)</cb:mo></cb:mrow></cb:math>, as long as boundary subregions are connected. This indicates that for holography, genuine multipartite entanglement is not small and plays an important role.