The black hole interior from non-isometric codes and complexity

The entanglement wedge reconstruction paradigm in AdS/CFT states that for a bulk qudit within the entanglement wedge of a boundary subregion $$ \overline{A} $$ A ¯ , operators acting on the bulk qudit can be reconstructed as CFT operators on $$ \overline{A} $$ A ¯ . This naturally fits within the framework of quantum error correction, with the CFT states containing the bulk qudit forming a code protected against the erasure of the boundary subregion A. In this paper, we set up and study a framework for relational bulk reconstruction in holography: given two code subspaces both protected against erasure of the boundary region A, the goal is to relate the operator reconstructions between the two spaces. To accomplish this, we assume that the two code subspaces are smoothly connected by a one-parameter family of codes all protected against the erasure of A, and that the maximally-entangled states on these codes are all full-rank. We argue that such code subspaces can naturally be constructed in holography in a “measurement-based” setting. In this setting, we derive a flow equation for the operator reconstruction of a fixed code subspace operator using modular theory which can, in principle, be integrated to relate the reconstructed operators all along the flow. We observe a striking resemblance between our formulas for relational bulk reconstruction and the infinite-time limit of Connes cocycle flow, and take some steps towards making this connection more rigorous. We also provide alternative derivations of our reconstruction formulas in terms of a canonical reconstruction map we call the modular reflection operator.

Relational bulk reconstruction from modular flow

Parrikar,

Rajgadia,

Singh

et al. 2024

Null states and time evolution in a toy model of black hole dynamics

Dong,

Kolanowski,

Liu

et al. 2024

Spacetime wormholes can provide non-perturbative contributions to the gravitational path integral that make the actual number of states eS in a gravitational system much smaller than the number of states $$ {e}^{S_{\textrm{p}}} $$ e S p predicted by perturbative semiclassical effective field theory. The effects on the physics of the system are naturally profound in contexts in which the perturbative description actively involves N = O(eS) of the possible $$ {e}^{S_{\textrm{p}}} $$ e S p perturbative states; e.g., in late stages of black hole evaporation. Such contexts are typically associated with the existence of non-trivial quantum extremal surfaces. However, by forcing a simple topological gravity model to evolve in time, we find that such effects can also have large impact for N ≪ eS (in which case no quantum extremal surfaces can arise). In particular, even for small N, the insertion of generic operators into the path integral can cause the non-perturbative time evolution to differ dramatically from perturbative expectations. On the other hand, this discrepancy is small for the special case where the inserted operators are non-trivial only in a subspace of dimension D ≪ eS. We thus study this latter case in detail. We also discuss potential implications for more realistic gravitational systems.

Constructing all BPS black hole microstates from the gravitational path integral

Boruch,

Iliesiu,

Yan

2024

Understanding how to prepare and count black hole micro-states by using the gravitational path integral is one of the most important problems in quantum gravity. Nevertheless, a state-by-state count of black hole microstates is difficult because the apparent number of degrees of freedom available in the gravitational effective theory can vastly exceed the entropy of the black hole, even in the special case of BPS black holes. In this paper, we show that we can use the gravitational path integral to prepare a basis for the Hilbert space of all BPS black hole microstates. We find that the dimension of this Hilbert space computed by an explicit state count is in complete agreement with the degeneracy obtained from the Gibbons-Hawking prescription. Specifically, this match includes all non-perturbative corrections in 1/GN. Such corrections are, in turn, necessary in order for this degeneracy of BPS states to match the non-perturbative terms in the 1/GN expansion in the string theory count of such microstates.