Quantum error correction codes and absolutely maximally entangled states

TL;DR

This paper develops methods to derive quantum error correction codes from absolutely maximally entangled states, analyzes their properties, and explores their implications for entanglement entropy and holographic models.

Contribution

It introduces a systematic approach to determine stabilizer generators and logical operators for codes derived from AME states, linking quantum error correction with entanglement structures.

Findings

Codes encode up to half the qudits in the original system.

Corrections to the Ryu-Takayanagi formula are identified for entangled inputs.

Entanglement entropy bounds are saturated by AME states.

Abstract

For every stabiliser $N$-qudit absolutely maximally entangled state, we present a method for determining the stabiliser generators and logical operators of a corresponding quantum error correction code. These codes encode $k$ qudits into $N-k$ qudits, with $k\leq \left \lfloor{N/2} \right \rfloor$, where the local dimension $d$ is prime. We use these methods to analyse the concatenation of such quantum codes and link this procedure to entanglement swapping. Using our techniques, we investigate the spread of quantum information on a tensor network code formerly used as a toy model for the AdS/CFT correspondence. In this network, we show how corrections arise to the Ryu-Takayanagi formula in the case of entangled input state, and that the bound on the entanglement entropy of the boundary state is saturated for absolutely maximally entangled input states.