# Quantum Codes of Maximal Distance and Highly Entangled Subspaces

## TL;DR

This paper establishes new bounds on quantum maximum distance separable (QMDS) codes, linking their existence to highly entangled subspaces, and confirms the quantum MDS conjecture for distance-three codes.

## Contribution

It generalizes bounds for QMDS codes, provides their weight distribution, and connects these codes to highly entangled subspaces, advancing understanding in quantum error correction.

## Key findings

- Bound on length of QMDS codes: n ≤ D^2 + d - 2
- Confirmed quantum MDS conjecture for distance-three codes
- Linked QMDS codes to highly entangled subspaces

## Abstract

We present new bounds on the existence of general quantum maximum distance separable codes (QMDS): the length $n$ of all QMDS codes with local dimension $D$ and distance $d \geq 3$ is bounded by $n \leq D^2 + d - 2$. We obtain their weight distribution and present additional bounds that arise from Rains' shadow inequalities. Our main result can be seen as a generalization of bounds that are known for the two special cases of stabilizer QMDS codes and absolutely maximally entangled states, and confirms the quantum MDS conjecture in the special case of distance-three codes. As the existence of QMDS codes is linked to that of highly entangled subspaces (in which every vector has uniform $r$-body marginals) of maximal dimension, our methods directly carry over to address questions in multipartite entanglement.