Spectral triples and Connes distances of qubits

We construct spectral triples of one- and two-qubit states and study the Connes spectral distances. We also construct the Dirac operator corresponding to the normal quantum trace distances. Based on the Connes spectral distances, we define a coherence measure of quantum states, and calculate the coherence of one-qubit states. We also study some simple cases about two-qubit states, and the corresponding spectral distances satisfy the Pythagoras theorem. These results are significant for the study of physical relations and geometric structures of qubits and other quantum states.