Bulk private curves require large conditional mutual information
Abstract We prove a theorem showing that the existence of “private” curves in the bulk of AdS implies two regions of the dual CFT share strong correlations. A private curve is a causal curve which avoids the entanglement wedge of a specified boundary region U $$ \mathcal{U} $$ . The implied correlat...
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Online Access: | https://doi.org/10.1007/JHEP09(2021)042 |
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doaj-e7b34cbf34414f069022a1718d2ccefd2021-09-12T12:02:23ZengSpringerOpenJournal of High Energy Physics1029-84792021-09-012021912810.1007/JHEP09(2021)042Bulk private curves require large conditional mutual informationAlex May0Department of Physics and Astronomy, The University of British ColumbiaAbstract We prove a theorem showing that the existence of “private” curves in the bulk of AdS implies two regions of the dual CFT share strong correlations. A private curve is a causal curve which avoids the entanglement wedge of a specified boundary region U $$ \mathcal{U} $$ . The implied correlation is measured by the conditional mutual information I V 1 : V 2 U $$ I\left({\mathcal{V}}_1:\left.{\mathcal{V}}_2\right|\mathcal{U}\right) $$ , which is O(1/G N ) when a private causal curve exists. The regions V 1 $$ {\mathcal{V}}_1 $$ and V 2 $$ {\mathcal{V}}_2 $$ are specified by the endpoints of the causal curve and the placement of the region U $$ \mathcal{U} $$ . This gives a causal perspective on the conditional mutual information in AdS/CFT, analogous to the causal perspective on the mutual information given by earlier work on the connected wedge theorem. We give an information theoretic argument for our theorem, along with a bulk geometric proof. In the geometric perspective, the theorem follows from the maximin formula and entanglement wedge nesting. In the information theoretic approach, the theorem follows from resource requirements for sending private messages over a public quantum channel.https://doi.org/10.1007/JHEP09(2021)042AdS-CFT CorrespondenceClassical Theories of Gravity |
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DOAJ |
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English |
format |
Article |
sources |
DOAJ |
author |
Alex May |
spellingShingle |
Alex May Bulk private curves require large conditional mutual information Journal of High Energy Physics AdS-CFT Correspondence Classical Theories of Gravity |
author_facet |
Alex May |
author_sort |
Alex May |
title |
Bulk private curves require large conditional mutual information |
title_short |
Bulk private curves require large conditional mutual information |
title_full |
Bulk private curves require large conditional mutual information |
title_fullStr |
Bulk private curves require large conditional mutual information |
title_full_unstemmed |
Bulk private curves require large conditional mutual information |
title_sort |
bulk private curves require large conditional mutual information |
publisher |
SpringerOpen |
series |
Journal of High Energy Physics |
issn |
1029-8479 |
publishDate |
2021-09-01 |
description |
Abstract We prove a theorem showing that the existence of “private” curves in the bulk of AdS implies two regions of the dual CFT share strong correlations. A private curve is a causal curve which avoids the entanglement wedge of a specified boundary region U $$ \mathcal{U} $$ . The implied correlation is measured by the conditional mutual information I V 1 : V 2 U $$ I\left({\mathcal{V}}_1:\left.{\mathcal{V}}_2\right|\mathcal{U}\right) $$ , which is O(1/G N ) when a private causal curve exists. The regions V 1 $$ {\mathcal{V}}_1 $$ and V 2 $$ {\mathcal{V}}_2 $$ are specified by the endpoints of the causal curve and the placement of the region U $$ \mathcal{U} $$ . This gives a causal perspective on the conditional mutual information in AdS/CFT, analogous to the causal perspective on the mutual information given by earlier work on the connected wedge theorem. We give an information theoretic argument for our theorem, along with a bulk geometric proof. In the geometric perspective, the theorem follows from the maximin formula and entanglement wedge nesting. In the information theoretic approach, the theorem follows from resource requirements for sending private messages over a public quantum channel. |
topic |
AdS-CFT Correspondence Classical Theories of Gravity |
url |
https://doi.org/10.1007/JHEP09(2021)042 |
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AT alexmay bulkprivatecurvesrequirelargeconditionalmutualinformation |
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1717755305398370304 |