Note on Connes spectral distances of qubits

Abstract: By virtue of the mathematical tools in noncommutative geometry, we study the Connes spectral distances between one- and two-qubit states. We construct a spectral triple corresponding to the $2D$ fermionic phase spaces, and calculate the Connes spectral distance between one qubits. Based on the Connes spectral distance, 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. We find that the Connes spectral distances are different from quantum trace distances. The Connes spectral distance can be considered as a significant supplement to the trace distances in quantum information sciences. These results are significant for the study of physical relations and geometric structures of qubits and other quantum states.