The Entropy Characterization of Quantum MDS Codes

TL;DR

This paper fully characterizes the entropy properties of quantum MDS codes, revealing that their joint states are pure and maximally mixed in specific subsystems, which deepens understanding of their quantum information structure.

Contribution

It provides a complete entropy characterization of quantum MDS codes, linking their code parameters to the entropic properties of their joint states.

Findings

Joint state of quantum MDS codes is pure.

Sub-systems up to half the total qudits are maximally mixed.

Entropy conditions precisely describe quantum MDS code structure.

Abstract

An $[[n,k,d]]$ quantum maximum-distance-separable code maps $k$ source qudits to $n$ coded qudits such that any $n-(d-1)$ coded qudits may recover all source qudits and $n = k + 2 (d-1)$. The entropy of the joint state of the reference system of $k$ qudits and the $n$ coded qudits is fully characterized - the joint state must be pure, i.e., has entropy zero; and any sub-system whose number of qudits is at most half of $k+n$, the total number of qudits in the joint state must be maximally mixed, i.e., has entropy equal to its size.