Classical codes in quantum state space

TL;DR

This paper introduces a method to construct quantum states and operators using classical coding theory, linking their distances to Hamming distances, and recovers the discrete Wigner function with applications in quantum information.

Contribution

It provides a novel construction connecting classical codes to quantum states and operators, enabling new applications in quantum information theory.

Findings

Construction relates quantum states to classical codes via Hamming distance

Recovers the discrete Wigner function using Reed-Solomon codes

Facilitates classical coding theory applications in quantum state arrangements

Abstract

We present a construction of Hermitian operators and quantum states labelled by strings from a finite field. The distance between these operators or states is then simply related (typically, proportional) to the Hamming distance between their corresponding strings. This allows a straightforward application of classical coding theory to find arrangements of operators or states with a given distance distribution. Using the simplex or extended Reed-Solomon code in our construction recovers the discrete Wigner function, which has important applications in quantum information theory.