Quantum Hamming Metrics

TL;DR

This paper develops a quantum version of the Hamming metric by extending classical concepts to non-commutative C*-algebras, providing a new framework for quantum metrics and their Kantorovich-Wasserstein counterparts.

Contribution

It introduces a novel approach to defining quantum Hamming metrics by expressing classical metrics in terms of C*-algebras and generalizing to non-commutative cases.

Findings

Quantum Hamming metric derived from classical metrics on function algebras.

Extension of quantum metrics to infinite-dimensional C*-algebras.

Framework for quantum Kantorovich-Wasserstein metrics on states.

Abstract

Given the set of words of a given length for a given alphabet, the Hamming metric between two such words is the number of positions where the two words differ. A quantum version of the corresponding Kantorovich-Wasserstein metric on states was introduced in 2021 by De Palma, Marvian, Trevisan and Lloyd. For the quantum version the alphabet is replaced by a full matrix algebra, and the set of words is replaced by the tensor product of a corresponding number of copies of that full matrix algebra. While De Palma et al. work primarily at the level of states, they do obtain the corresponding seminorm (the quantum Hamming metric) on the algebra of observables that plays the role of assigning Lipschitz constants to functions. A suitable such seminorm on a unital C*-algebra is the current common method for defining a quantum metric on a C*-algebra. In this paper we will reverse the process, by first expressing the Hamming metric in terms of the C*-algebra of functions on the set of words, and then dropping the requirement that the algebra be commutative so as to obtain the quantum Hamming metric. From that we obtain the corresponding Kantorovich-Wasserstein metric on states. Along the way we show that many of the steps can be put in more general forms of some interest, notably for infinite-dimensional C*-algebras.