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Connes spectral distances, quantum discord and coherence of qubits

Bing-Sheng Lin, Zi-Hao Xu, Ji-Hong Wang, Han-Liang Chen

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.

Connes spectral distances, quantum discord and coherence of qubits

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.
Paper Structure (7 sections, 5 theorems, 106 equations, 2 figures)

This paper contains 7 sections, 5 theorems, 106 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $2D$ fermionic phase space and spectral triple
  3. Connes spectral distances between one-qubit states
  4. Connes spectral distance measure of quantum discord and quantum coherence
  5. The Dirac operators for Euclidean distances and quantum trace distances
  6. Connes spectral distances between two-qubit states
  7. Discussions and conclusions

Key Result

Theorem 1

In the spectral triple $(\mathcal{A},\mathcal{H},\mathcal{D})$, the Connes spectral distance between one-qubit states $\rho_1$ and $\rho_2$ is

Figures (2)

  • Figure 1: Connes spectral distances between one-qubit states in the Bloch sphere.
  • Figure 2: Connes spectral distances between two-qubit states $|ij \rangle$.

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Definition 3
  • Definition 4
  • Lemma 1
  • Theorem 2
  • Corollary 1
  • Definition 5
  • ...and 1 more