Unitary dynamics and resource trade-offs in a four-qubit isotropic Heisenberg XXX chain with tunable next-nearest-neighbor coupling

Abstract

This study derives the unitary dynamics of a four qubit Heisenberg XXX chain with tunable next nearest neighbor coupling $\alpha$, starting from a Bell type initial state, and analyzes the evolution of quantum resources under the phase $\phi = (\alpha + 1)t$. We provide closed form expressions for fidelity $F(\rho(0),\rho(t))$, coherence $C_{l_1}(\rho(t))$, and two qubit entanglement of formation $E_F(t)$ for subsystems $12$ and $34$, all of which are governed by $\phi$. Fidelity exhibits periodic behavior with $F = \lvert \cos(\phi/2) \rvert$ and a frozen regime at $\alpha = -1$ where $F \equiv 1$. Coherence follows $C_{l_1}(\rho(t)) = \sin^2(\phi/2)$, showing increasing sensitivity with $\lvert \alpha + 1 \rvert$ and vanishing at $\alpha = -1$. Entanglement of formation $E_F(t)$ is an entropic function of $\phi$, displaying banded oscillations and freezing at $\alpha = -1$. The phase $\phi$ unifies the behavior of all diagnostics, linking faster dynamics to larger $\lvert \alpha + 1 \rvert$ and revealing maximal sensitivity at $(\alpha + 1)t = \pi/4 + k\pi/2$. This integrated framework provides exact benchmarks for small quantum devices and a clear pathway to noise, finite temperature, and larger system extensions.