Emergent Mixed States for Baby Universes and Black Holes

TL;DR

This work analyzes how sequences of high-energy states in AdS/CFT behave in the large-$N$ limit when the bulk contains baby universes or black holes. It shows that pure-state sequences often fail to converge and instead approach mixed states, with gravitational wormholes driving variance in amplitudes; averaging over $N$ can yield a well-defined mixed-state limit whose GNS construction features a nontrivial commutant that may encode baby-universe degrees of freedom. By connecting toy qubit models to gravitational path integrals, the authors elucidate when the large-$N$ limit preserves pure-state information and when it does not, and discuss possible resolutions via $N$-averaging or operator-choice averaging. The results illuminate how wormhole physics induces nonlocal effects and probe the interpretation of baby universes within a controlled large-$N$ framework, offering insight into the emergence of mixed-state descriptions in holographic settings.

Abstract

We examine the behavior of sequences of states in the large $N$ limit of AdS/CFT duality in cases in which the bulk duals involve baby universes or black holes. Such sequences generally fail to converge as pure states. Under suitable conditions, such as diverging coarse-grained entropy, they can converge to mixed states for the large $N$ algebra, as in the case of black holes. For Euclidean preparations that produce baby universes, the sequences do not converge, due to wormhole contributions, and so these states cannot admit large $N$ limits. Nevertheless, appropriate averaging over $N$ can lead to convergence to a mixed state. The associated algebras have nontrivial commutants, which can possibly be interpreted as operators in the baby universe.

Figures (6)

• Figure 1: Left: At finite $N$, the high-energy state is pure and then dual to the entire bulk, whose geometry has quantum fluctuations and so there is no sharp horizon. Right: The limit of a sequence of high-energy pure states is dual to the mixed state in the gray bulk region.
• Figure 2: Left: the path integral that computes the norm of the state when no heavy operator is inserted. Cutting on the horizontal ellipses and continuing to Lorentzian signature leads to thermal AdS if the temperature is sufficiently low. Right: the path integral with ${\Bbb O}_N$ and its adjoint inserted (red ellipses). The gravitational saddle is proposed to include a baby universe (dashed line) supported by a matter shell (red line).
• Figure 3: The CFT path integral for $\widehat{M}_{ij}$. The blue dots represent insertions of operators in the single-trace algebra.
• Figure 4: The path integral that gives the variance of the norm of $\widehat{\Psi}_N$. The green surfaces are identified and the double line surfaces are identified. This is the same path integral that computes the second Rényi entropy.
• Figure 5: The two equal wormhole contributions to $\overline{|M_{ij}|^4}$. Only the left diagram contributes to $\overline{|M_{ij}|^2}^2$. We have suppressed the operator insertions in the diagram for visual ease.