# Constructing optimal quantum error correcting codes from absolute maximally entangled states

## TL;DR

This paper explores the connection between absolutely maximally entangled states and quantum error correcting codes, providing explicit constructions and a conjecture that leads to optimal codes saturating the quantum Singleton bound.

## Contribution

It establishes a link between AME states and classical MDS codes, introduces a stabilizer formalism for AME states, and constructs new QECCs based on these states.

## Key findings

- Explicit formulas for AME states with prime power dimensions
- A generalized Bell basis of AME states
- Construction of QECCs saturating the quantum Singleton bound

## Abstract

Absolutely maximally entangled (AME) states are pure multi-partite generalizations of the bipartite maximally entangled states with the property that all reduced states of at most half the system size are in the maximally mixed state. AME states are of interest for multipartite teleportation and quantum secret sharing and have recently found new applications in the context of high-energy physics in toy models realizing the AdS/CFT-correspondence. We work out in detail the connection between AME states of minimal support and classical maximum distance separable (MDS) error correcting codes and, in particular, provide explicit closed form expressions for AME states of $n$ parties with local dimension $q$ a power of a prime for all $q \geq n-1$. Building on this, we construct a generalization of the Bell-basis consisting of AME states and develop a stabilizer formalism for AME states. For every $q \geq n-1$ prime we show how to construct QECCs that encode a logical qudit into a subspace spanned by AME states. Under a conjecture for which we provide numerical evidence, this construction produces a family of quantum error correcting codes $[\![n,1,n/2]\!]_q$ for $n$ even, saturating the quantum Singleton bound. We show that our conjecture is equivalent to the existence of an operator whose support cannot be decreased by multiplying it with stabilizer products and explicitly construct the codes up to $n = 8$.