Quantum Wasserstein isometries on the qubit state space

TL;DR

This paper characterizes the isometries of the quantum bit state space under Wasserstein distances, revealing classical symmetries and novel non-surjective, non-injective isometries linked to specific cost operators.

Contribution

It provides a detailed description of Wasserstein isometries for qubits, including classical symmetry groups and new types of isometries for particular cost operators.

Findings

Isometry group with Pauli matrices is unitary/anti-unitary conjugations.

In the Bloch sphere, isometries match classical O(3) symmetries.

Discovery of non-surjective, non-injective isometries for clock and shift costs.

Abstract

We describe Wasserstein isometries of the quantum bit state space with respect to distinguished cost operators. We derive a Wigner-type result for the cost operator involving all the Pauli matrices: in this case, the isometry group consists of unitary or anti-unitary conjugations. In the Bloch sphere model, this means that the isometry group coincides with the classical symmetry group $\mathbf{O}(3).$ On the other hand, for the cost generated by the qubit "clock" and "shift" operators, we discovered non-surjective and non-injective isometries as well, beyond the regular ones. This phenomenon mirrors certain surprising properties of the quantum Wasserstein distance.