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

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 $F=\mid \cos (\phi \mathrm{/}2)\mid$ and a frozen regime at $\alpha =-1$ where $F\equiv 1$. Coherence follows $C_{l_1}(\rho (t))={\sin }^2(\phi \mathrm{/}2)$, showing increasing sensitivity with $\mid \alpha +1\mid$ 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 $\mid \alpha +1\mid$ and revealing maximal sensitivity at $(\alpha +1)t=\pi \mathrm{/}4+k\pi \mathrm{/}2$. This integrated framework provides exact benchmarks for small quantum devices and a clear pathway to noise, finite temperature, and larger system extensions.