Connes spectral distances, quantum discord and coherence of qubits

TL;DR

This paper develops spectral triples for qubits using the Hilbert-Schmidt formulation, introduces spectral distances, and proposes new measures for quantum discord and coherence, with explicit calculations for one- and two-qubit states.

Contribution

It constructs spectral triples and spectral distances for qubits, and introduces novel definitions of quantum discord and coherence based on Connes spectral distances.

Findings

Spectral distances satisfy the Pythagoras theorem for two-qubit states.

Explicit coherence calculations for one-qubit states.

Spectral distances provide insights into quantum state geometry.

Abstract

We construct spectral triples of one- and two-qubit states using the Hilbert-Schmidt operatorial formulation, 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 propose some definitions of quantum discord and coherence measure of quantum states, and explicitly 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 studies on physical relations and geometric structures of qubits and other quantum states.