Superadditivity in large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>N</mml:mi></mml:math> field theories and performance of quantum tasks

Field theories exhibit dramatic changes in the structure of their operator algebras in the limit where the number of local degrees of freedom (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>N</a:mi></a:math>) becomes infinite. An important example of this is that the algebras associated to local subregions may not be additively generated in the limit. We investigate examples and explore the consequences of this “superadditivity” phenomenon in large <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>N</c:mi></c:math> field theories and holographic systems. In holographic examples we find cases in which superadditive algebras can probe the black hole interior, while the additive algebra cannot. We also discuss how superaddivity explains the success of quantum error correction models of holography. Finally we demonstrate how superadditivity is intimately related to the ability of holographic field theories to perform quantum tasks that would naively be impossible. We argue that the connected wedge theorems (CWTs) of May, Penington, Sorce, and Yoshida, which characterize holographic protocols for quantum tasks, can be rephrased in terms of superadditive algebras and use this rephrasing to conjecture a generalization of the CWTs that is an equivalence statement.