Progress on Problem about Quantum Hamming Bound for Impure Quantum Codes

TL;DR

This paper investigates whether impure quantum error-correcting codes can violate the quantum Hamming bound, providing partial solutions and identifying thresholds where codes must adhere to the bound.

Contribution

It establishes a threshold condition under which all quantum codes must satisfy the quantum Hamming bound, advancing understanding of impure code limitations.

Findings

Existence of a threshold for code parameters

Impure codes obey the quantum Hamming bound below the threshold

Provided specific values for small code distances and binary codes

Abstract

A famous open problem in the theory of quantum error-correcting codes is whether or not the parameters of an impure quantum code can violate the quantum Hamming bound for pure quantum codes. We partially solve this problem. We demonstrate that there exists a threshold such that an arbitrary quantum code must obey the quantum Hamming bound whenever . We list some values of for small d and binary quantum codes.