Superadditivity in large $N$ field theories and performance of quantum tasks

TL;DR

We show that in the large $N$ limit, local operator algebras can lose additivity and exhibit superadditivity, where ${\mathcal{X}}_{R_1} \lor {\mathcal{X}}_{R_2} \subsetneq {\mathcal{X}}_{R_1 \cup R_2}$, enabling nonlocal boundary descriptions of bulk physics. In holographic CFTs, this algebraic nonlocality underpins entanglement wedge nesting and interior access via joint boundary regions, while additive algebras may fail to probe behind horizons. The work connects superadditivity to quantum error correction in holography, and reformulates holographic quantum tasks through a generalized connected wedge theorem (GCWT), positing an equivalence between nonlocally generated boundary operators and the feasibility of nonlocal boundary protocols. It provides concrete $d=2$ and higher-dimensional examples, analyzes finite-$N$ realizations through approximate code subspaces and modular reconstruction, and outlines open questions about the precise nature of nonlocal operators and their finite-$N$ behavior.

Abstract

Field theories exhibit dramatic changes in the structure of their operator algebras in the limit where the number of local degrees of freedom ($N$) 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 $N$ 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 sucess 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 naievely 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 re-phrased in terms of superadditive algebras and use this re-phrasing to conjecture a generalization of the CWTs that is an equivalence statement.

Figures (18)

• Figure 1: About the vacuum state of a CFT$_2$ the algebra ${\mathcal{X}}_{R_-}\vee{\mathcal{X}}_{R_+}$ is equal to the algebra generated by single-trace operators in the blue region depicted in (a) (see Sec. \ref{['sec:2din']} for more detail). In contrast, ${\mathcal{X}}_{R_- \cup R_+}$ is equal to the algebra generated by single-trace operators in the entire green region of (b). Clearly ${\mathcal{X}}_{R_-} \lor {\mathcal{X}}_{R_+} \subsetneq {\mathcal{X}}_{R_- \cup R_+}$.
• Figure 2: $x^0=0$ slice of Poincare AdS$_3$ is shown in the plots. The proper inclusion ${\mathfrak{b}}_{R_-}\cup{\mathfrak{b}}_{R_+} \subsetneq {\mathfrak{b}}_{R_- \cup R_+}$ is the bulk geometric realization of the superadditivity ${\mathcal{X}}_{R_-}\vee{\mathcal{X}}_{R_+} \subsetneq {\mathcal{X}}_{R_- \cup R_+}.$
• Figure 3: The $x^0=0$ slice of Poincaré AdS. Extra operators in ${\mathcal{X}}_{R_- \cup R_+}$ that are not in ${\mathcal{X}}_{R_-}\vee{\mathcal{X}}_{R_+}$ are realized geometrically in the bulk as operators supported in the green shaded region.
• Figure 4: The superadditivity $\vee_{i=1}^{3} {\mathcal{X}}_{R_i} \subsetneq {\mathcal{X}}_{B}$ is geometrically realized in the bulk by $\cup_{i=1}^{3} {\mathfrak{b}}_{R_i}$ (the union of the regions between the dashed blue lines and the boundary) being a proper subset of the full bulk Cauchy slice $\Sigma$ (the entire green disk).
• Figure 5: Bulk subregions on the $t=0$ slice of the dual at finite temperature. The duals of ${\mathcal{X}}_{R_-} \vee {\mathcal{X}}_{R_+}$ and ${\mathcal{X}}_{R_-\cup R_+}$ are, respectively, the blue and the union of blue and green shaded regions in (a). The mutual information is computed holographically as the sum of areas of the blue and magenta surfaces minus that of the green surface in (b). In the high temperature limit, this is equal to the area of the RT surface for the region $R_+ \cap R_-.$