Quantum correlations versus spin magnitude: Transition to the classical limit

TL;DR

The paper investigates how quantum correlations in a spin-$j$ and spin-$\tfrac{1}{2}$ system transition toward classical behavior as $j$ grows, using closed-form expressions for Local Quantum Uncertainty (LQU) and Local Quantum Fisher Information (LQFI) under axial symmetry. It shows that entanglement vanishes in the classical limit with a threshold temperature $T_{\rm th}$ that satisfies $T_{\rm th}=\frac{(2j+1)J_0}{2\sqrt{j(j+1)}\ln(2j+2)}$ and tends to zero as $j\to\infty$, while SU(2)-invariant ground-state discord-like correlations (LQU/LQFI) persist and approach $2/3$ as $j\to\infty$ but are unstable to symmetry-breaking perturbations. The XXZ model and non-uniform field perturbations reveal that anisotropy and external fields remove oscillations and stabilize the correlations, illustrating the fragility of high-symmetry quantum features in the classical limit. The work provides exact analytic formulas for LQU/LQFI, elucidating the mechanisms of quantum-to-classical transitions in macroscopic spins and highlighting persistent yet fragile quantum correlations beyond entanglement in highly symmetric states.

Abstract

Quantum-classical transitions have long attracted much attention. We study such transitions in quantum spin-($j$,1/2) systems at thermal equilibrium. Unlike the previous work [Phys. Rev. A 73, 064302 (2006)], it is found that the threshold temperature of quantum entanglement decreases with increasing spin $j$ and completely disappears in the limit $j\to\infty$. In the ground state of systems with highly symmetric interactions, the discord-type quantum correlations can exist even for arbitrarily large spin. Such correlations turn out to be unstable and are destroyed by small perturbations that violate the symmetry of the Hamiltonian. The stable quantum correlations gradually degrade as the spin $j$ grows and eventually vanish when the classical limit is reached.

Figures (10)

• Figure 1: (Color online) Local quantum Fisher information $\cal F$ ($\rm\bf a$) and local quantum uncertainty $\cal U$ ($\rm\bf b$) versus reduced temperature $T/J_0$ for the antiferromagnetic Heisenberg systems $(j,1/2)$ with $j=1$ (blue), $j=3/2$ (green), $j=2$ (orange) and $j=105/2$ (red).
• Figure 2: (Color online) Local quantum Fisher information $\cal F$ ($\rm\bf a$) and local quantum uncertainty $\cal U$ ($\rm\bf b$) versus reduced temperature $T/|J_0|$ for the ferromagnetic Heisenberg systems $(j,1/2)$ with $j=1$ (blue), $j=3/2$ (green), $j=2$ (orange) and $j=105/2$ (red).
• Figure 3: (Color online) Local quantum Fisher information as a function of $F$ for the different spin angular momentum quantum numbers $j=1$ (blue), $j=3/2$ (green), $j=2$ (orange) and $j=105/2$ (red). Points $F=0$ and $F=1$ correspond to the zero temperature $T=0$ for a ferromagnet and an antiferromagnet, respectively. The local minima at $F=j/(2j+1)$ correspond to the infinitely high temperatures.
• Figure 4: (Color online) Normalized threshold temperature $T_{\rm th}/J_0$ as a function of spin parameter $j$.
• Figure 5: (Color online) Ground-state quantum correlations vs $j$ in the XXX system: LQFI/LQU for $J_0>0$ (red line), LQFI/LQU for $J_0<0$ (blue line), black dashed horizontal line corresponds to their asymptotic value 2/3, EoF (orange dashed line) and double negativity $\cal N$ (brown dashed line).