Non-Isometric Quantum Error Correction in Gravity

TL;DR

This work models non-isometric quantum error correction in a gravity-inspired setting by constructing a holographic dictionary V as a complex Gaussian random map between a microcanonical bulk code space and the boundary theory. Using measure concentration, it proves that overlaps and norms are preserved for a subexponential-sized set S of code states, enabling state-specific reconstructions of code operators in line with entanglement wedge expectations. The results illuminate how bulk effective field theory can emerge and fail within a non-isometric encoding, depending on the entanglement structure and state class, and they connect gravitational path integrals with QECC notions, including an equilibrium basis and relative notions of complexity. Overall, the paper provides a tractable framework linking Euclidean gravity, non-isometric QECCs, and bulk reconstruction, with implications for black hole information, complexity, and EFT breakdowns in quantum gravity.

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.

Figures (2)

• Figure 1: (Left) The disk contribution to the bulk Euclidean path integral computing the thermal partition function. The boundary has renormalized length $\beta$. Cutting the path integral along the dashed blue line gives a state in $\mathcal{H}_\text{grav}$, the Hilbert space of dilaton gravity on an interval with two asymptotic boundaries. This state is the eternal two-sided black hole in AdS$_2$. (Right) The boundary Euclidean path integral computing the thermal partition function. Cutting the path integral along the dashed blue line gives the thermofield double state in two copies $\mathcal{H}_B \otimes \mathcal{H}_B$ of the microscopic Hilbert space. In the asymptotic $e^{-S_0}$ expansion of JT gravity \ref{['eq:jt-action']}, this path integral is computed by summing over all possible Euclidean geometries and topologies which fill in the circle. This includes the disk, the disk with a handle, and so on.
• Figure 2: (Left) The disk contribution to the bulk Euclidean path integral computing the one-brane (red curve) partition function. The boundary has renormalized length $\beta$. Cutting the path integral along the dashed blue line gives a state in $\mathcal{H}_\text{br}$, the Hilbert space of dilaton gravity on an interval with one asymptotic boundary and one brane boundary. (Right) The boundary Euclidean path integral computing the one-brane partition function. Cutting the path integral along the dashed blue line gives the projected thermofield double state $\langle S_\mu \vert \text{TFD}(\beta) \rangle$, which we call $\vert {\text{BR}}(\beta) \rangle$ in \ref{['eq:BR']}, in the microscopic Hilbert space $\mathcal{H}_B$. The red dots are brane boundary conditions.