Non-Isometric Quantum Error Correction in Gravity

TL;DR

This paper introduces non-isometric quantum error correcting codes in a black hole model, showing they can preserve inner products for large state sets and enabling state-specific reconstructions, with implications for bulk effective field theory.

Contribution

It constructs and analyzes non-isometric error correcting codes in a black hole toy model, demonstrating their ability to preserve inner products and facilitate state-specific reconstructions.

Findings

Typical codes preserve pairwise inner products in large state sets.

Existence of state-specific reconstructions for certain subspaces.

Upper bounds on bulk effective field theory Hilbert space size.

Abstract

We construct and study an ensemble of non-isometric error correcting codes in a toy model of an evaporating black hole in two-dimensional dilaton gravity. In the preferred bases of Euclidean path integral states in the bulk and Hamiltonian eigenstates in the boundary, the encoding map is proportional to a linear transformation with independent complex Gaussian random entries of zero mean and unit variance. Using measure concentration, we show that the typical such code is very likely to preserve pairwise inner products in a set $S$ of states that can be subexponentially large in the microcanonical Hilbert space dimension of the black hole. The size of this set also serves as an upper limit on the bulk effective field theory Hilbert space dimension. Similar techniques are used to demonstrate the existence of state-specific reconstructions of $S$-preserving code space unitary operators. State-specific reconstructions on subspaces exist when they are expected to by entanglement wedge reconstruction. We comment on relations to complexity theory and the breakdown of bulk effective field theory.