More on genuine multientropy and holography
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Abstract
<jats:p>By generalizing the construction of genuine multientropy <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msup><a:mrow><a:mi>GM</a:mi></a:mrow><a:mrow><a:mo stretchy="false">(</a:mo><a:mi mathvariant="monospace">q</a:mi><a:mo stretchy="false">)</a:mo></a:mrow></a:msup></a:mrow></a:math> for genuine multipartite entanglement proposed in the previous paper [Genuine multi-entropy and holography, .], we give a prescription on how to construct <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mrow><f:msup><f:mrow><f:mi>GM</f:mi></f:mrow><f:mrow><f:mo stretchy="false">(</f:mo><f:mi mathvariant="monospace">q</f:mi><f:mo stretchy="false">)</f:mo></f:mrow></f:msup></f:mrow></f:math> systematically for any <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi mathvariant="monospace">q</k:mi></k:math>. The crucial point is that our construction naturally fits to the partition number <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi>p</n:mi><n:mo stretchy="false">(</n:mo><n:mi mathvariant="monospace">a</n:mi><n:mo stretchy="false">)</n:mo></n:math> of integer <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi mathvariant="monospace">a</s:mi></s:math>. For general <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:mi mathvariant="monospace">q</v:mi></v:math>, <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mrow><y:msup><y:mrow><y:mi>GM</y:mi></y:mrow><y:mrow><y:mo stretchy="false">(</y:mo><y:mi mathvariant="monospace">q</y:mi><y:mo stretchy="false">)</y:mo></y:mrow></y:msup></y:mrow></y:math> contains <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:mi>N</db:mi><db:mo stretchy="false">(</db:mo><db:mi mathvariant="monospace">q</db:mi><db:mo stretchy="false">)</db:mo><db:mo>=</db:mo><db:mi>p</db:mi><db:mo stretchy="false">(</db:mo><db:mi mathvariant="monospace">q</db:mi><db:mo stretchy="false">)</db:mo><db:mo>−</db:mo><db:mi>p</db:mi><db:mo stretchy="false">(</db:mo><db:mi mathvariant="monospace">q</db:mi><db:mo>−</db:mo><db:mn>1</db:mn><db:mo stretchy="false">)</db:mo><db:mo>−</db:mo><db:mn>1</db:mn></db:math> number of free parameters. Furthermore, these give <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:mi>N</ob:mi><ob:mo stretchy="false">(</ob:mo><ob:mi mathvariant="monospace">q</ob:mi><ob:mo stretchy="false">)</ob:mo><ob:mo>+</ob:mo><ob:mn>1</ob:mn></ob:math> number of new diagnostics for genuine <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"><tb:mi mathvariant="monospace">q</tb:mi></tb:math>-partite entanglement. Especially for <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"><wb:mi mathvariant="monospace">q</wb:mi><wb:mo>=</wb:mo><wb:mn>4</wb:mn></wb:math> case, this reproduces not only the known diagnostics pointed out by Balasubramanian [Multiboundary wormholes and holographic entanglement, .], but also a new diagnostics for quadripartite entanglement. We also study these <zb:math xmlns:zb="http://www.w3.org/1998/Math/MathML" display="inline"><zb:mrow><zb:msup><zb:mrow><zb:mi>GM</zb:mi></zb:mrow><zb:mrow><zb:mo stretchy="false">(</zb:mo><zb:mi mathvariant="monospace">q</zb:mi><zb:mo stretchy="false">)</zb:mo></zb:mrow></zb:msup></zb:mrow></zb:math> for <ec:math xmlns:ec="http://www.w3.org/1998/Math/MathML" display="inline"><ec:mi mathvariant="monospace">q</ec:mi><ec:mo>=</ec:mo><ec:mn>4</ec:mn></ec:math>, 5 in holography and show that these are of the order of <hc:math xmlns:hc="http://www.w3.org/1998/Math/MathML" display="inline"><hc:mi mathvariant="script">O</hc:mi><hc:mrow><hc:mo stretchy="false">(</hc:mo><hc:mn>1</hc:mn><hc:mo>/</hc:mo><hc:msub><hc:mi>G</hc:mi><hc:mi>N</hc:mi></hc:msub><hc:mo stretchy="false">)</hc:mo></hc:mrow></hc:math> both analytically and numerically. Our results give evidence that genuine multipartite entanglement is ubiquitous in holography. We discuss the connection to quantum error correction and the role of genuine multipartite entanglement in bulk reconstruction.</jats:p>