Variance-based uncertainty relations and entanglement amplification for particles constrained on a torus

Abstract

We formulate the variance-based uncertainty relations (URs) via the Robertson’s inequality, for a 2-particle entangled system constrained on a torus and subject to a stationary magnetic field $\overrightarrow{\mathcal{B}}$. We explore the system’s parameter space and show that these new URs have field-tunable uncertainty bounds. Our analysis reveals that $\overrightarrow{\mathcal{A}}$ (vector potential) induces a phase shift in the state, due to the Aharonov–Bohm effect, leading to a perturbed system dynamics which results in asymmetric product of variance ($\mathcal{POV}$). Additionally, we give the critical range of $\overrightarrow{\mathcal{A}}$ and $\overrightarrow{\mathcal{B}}$ where the system acts as an entanglement amplifier; this amplification is also discussed under various geometric parameters. The possibility of reducing the $\mathcal{POV}$ of the conjugate pair [q, p] below the known benchmark value by the Generalized Uncertainty Relation (GUR) is also demonstrated. Finally, we check the susceptibility of state coherence to $\overrightarrow{\mathcal{B}}$ by saturating the angular momentum uncertainty relation and identify the critical coherence value ${\mathcal{B}}_c$ such that when $\mathcal{B}\ne {\mathcal{B}}_c$, the state decoheres.