Quantum Wasserstein isometries on the qubit state space

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