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

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 B⃗ B → . We explore the system’s parameter space and show that these new URs have field-tunable uncertainty bounds. Our analysis reveals that A⃗ 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 ( POV POV ). Additionally, we give the critical range of A⃗ A → and B⃗ B → where the system acts as an entanglement amplifier; this amplification is also discussed under various geometric parameters. The possibility of reducing the POV 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 B⃗ B → by saturating the angular momentum uncertainty relation and identify the critical coherence value Bc B c such that when B≠ Bc B ≠ B c , the state decoheres.