Quantum Hamming Metrics

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.