Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonclassicality Analysis and Entanglement Witnessing in Spin-$1/2$ NMR Systems

Fatemeh Khashami

TL;DR

This work analyzes nonclassical features in thermally equilibrated two-spin-$1/2$ NMR systems under external fields by deriving closed-form expressions for coherence and mixedness and by introducing an experimentally accessible entanglement witness. It shows a quantum critical point at zero temperature through nonanalytic behavior in coherence and a peak in mixedness at a field-induced level crossing, while connecting these quantum-information quantifiers to observable NMR spectra. The authors provide an experimental protocol to estimate mixedness from standard polarization measurements and demonstrate how entanglement can be certified via a singlet-witness based on measurable spin correlators. Overall, the study establishes a practical bridge between quantum information concepts and NMR spectroscopy, enabling simultaneous assessment of coherence, purity, and entanglement in coupled spin systems and informing quantum metrology and thermometry in spectroscopic platforms.

Abstract

We investigate quantum features and non-classical nature of two-spin-$1/2$ NMR systems at thermal equilibrium under external magnetic fields. More specifically, using suitable quantifiers, we analyze quantum coherence, mixedness, and entanglement in NMR systems and examine their features within the system. We derive closed-form analytical expressions for the quantum elements and show how they depend on temperature and magnetic field strength. We demonstrate that at zero temperature, the system exhibits a quantum critical point, characterized by non-analytic behavior in the measures of coherence, and a sharp peak in mixedness. Moreover, we analyze the entanglement of the system using a suitable entanglement witness. This provides an experimentally friendly setting for testing entanglement in NMR systems. In other words, the witness links the entanglement in the system to quantum observables, making it directly provable in NMR experiments. We establish a connection between quantum information quantifiers and experimentally accessible NMR spectra of the system, enabling the quantification of entanglement, coherence, and mixedness through NMR signal processing.

Nonclassicality Analysis and Entanglement Witnessing in Spin-$1/2$ NMR Systems

TL;DR

This work analyzes nonclassical features in thermally equilibrated two-spin- NMR systems under external fields by deriving closed-form expressions for coherence and mixedness and by introducing an experimentally accessible entanglement witness. It shows a quantum critical point at zero temperature through nonanalytic behavior in coherence and a peak in mixedness at a field-induced level crossing, while connecting these quantum-information quantifiers to observable NMR spectra. The authors provide an experimental protocol to estimate mixedness from standard polarization measurements and demonstrate how entanglement can be certified via a singlet-witness based on measurable spin correlators. Overall, the study establishes a practical bridge between quantum information concepts and NMR spectroscopy, enabling simultaneous assessment of coherence, purity, and entanglement in coupled spin systems and informing quantum metrology and thermometry in spectroscopic platforms.

Abstract

We investigate quantum features and non-classical nature of two-spin- NMR systems at thermal equilibrium under external magnetic fields. More specifically, using suitable quantifiers, we analyze quantum coherence, mixedness, and entanglement in NMR systems and examine their features within the system. We derive closed-form analytical expressions for the quantum elements and show how they depend on temperature and magnetic field strength. We demonstrate that at zero temperature, the system exhibits a quantum critical point, characterized by non-analytic behavior in the measures of coherence, and a sharp peak in mixedness. Moreover, we analyze the entanglement of the system using a suitable entanglement witness. This provides an experimentally friendly setting for testing entanglement in NMR systems. In other words, the witness links the entanglement in the system to quantum observables, making it directly provable in NMR experiments. We establish a connection between quantum information quantifiers and experimentally accessible NMR spectra of the system, enabling the quantification of entanglement, coherence, and mixedness through NMR signal processing.

Paper Structure

This paper contains 13 sections, 33 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Methods
  3. Hamiltonian Description of a Thermally Polarized Two-Spin-1/2 NMR System
  4. Thermal State Representation of the Two-Spin-$1/2$ NMR System
  5. Field-Dependent Level Dynamics and Crossing Behavior
  6. Results
  7. Relative Entropy of Coherence For the Two-Spin-$1/2$ NMR System
  8. Mixedness For the Two-Spin-$1/2$ NMR System
  9. NMR Signal Simulation based on Mixedness Analysis Near the Critical Point
  10. Experimental Protocol For Estimation of Mixedness in Two-Spin-$1/2$ NMR Systems
  11. Entanglement Detection via Witness Operators in Two-Spin-$1/2$ NMR Systems
  12. Discussions
  13. Conclusion

Figures (8)

  • Figure 1: (a) NMR spectrum for two-spin-$1/2$. (b) Field-dependent behavior of a homonuclear system, magnetic field strength changes from zero-field to low-field and finally reaches high-field. A level crossing occurs between $\mathrm{E}_3$ and $\mathrm{E}_4$, indicating a potential quantum transition point.
  • Figure 2: Relative entropy of coherence, $\mathcal{R}(\rho)$, versus the rescaled temperature for different values of the dimensionless magnetic field parameter $\omega_{\Sigma}/{\boldsymbol{J}}$. Panels (a), (b), and (c) correspond to $\omega_{\delta}/{\boldsymbol{J}} = 0$ (a homonuclear system), $\omega_{\delta}/{\boldsymbol{J}} = 1$ (a heteronuclear system), and $\omega_{\delta}/{\boldsymbol{J}} = 2.5$ (a heteronuclear system), respectively. The re-scaled temperature parameter is $\tau = k_B T /{\boldsymbol{J}}$.
  • Figure 3: Temperature dependence of $\mathcal{R}(\rho)$ for different values of the dimensionless magnetic field parameter $\omega_{\Sigma}/{\boldsymbol{J}}$. Panels (a), (b), and (c) correspond to $\omega_{\delta}/{\boldsymbol{J}} = 0$ (a homonuclear system), $\omega_{\delta}/{\boldsymbol{J}} = 1$ (a heteronuclear system), and $\omega_{\delta}/{\boldsymbol{J}} = 2.5$ (a heteronuclear system), respectively.
  • Figure 4: Mixedness as a function of the rescaled temperature for different values of the dimensionless magnetic field parameter $\omega_{\Sigma}/{\boldsymbol{J}}$. Panels (a), (b), and (c) correspond to $\omega_{\delta}/{\boldsymbol{J}} = 0$ (a homonuclear system), $\omega_{\delta}/{\boldsymbol{J}} = 1$ (a heteronuclear system), and $\omega_{\delta}/{\boldsymbol{J}} = 2.5$ (a heteronuclear system), respectively. The re-scaled temperature parameter is $\tau = k_B T /{\boldsymbol{J}}$.
  • Figure 5: Mixedness as a function of the normalized frequency ratio $\omega_\Sigma / {\boldsymbol{J}}$ for varying values of the re-scaled temperature. Panels (a), (b), and (c) correspond to $\omega_{\delta}/{\boldsymbol{J}} = 0$ (a homonuclear system), $\omega_{\delta}/{\boldsymbol{J}} = 1$ (a heteronuclear system), and $\omega_{\delta}/{\boldsymbol{J}} = 2.5$ (a heteronuclear system), respectively, illustrating the influence of magnetic field strength and detuning on thermal mixedness.
  • ...and 3 more figures