Hamiltonian-Generated Dynamics and Quantum Resource Allocation in a 4-Qubit Isotropic Heisenberg Spin Ring with Adjustable Next-Nearest-Neighbor Interactions

Abstract

This investigation delivers a fully analytical solution for the unitary evolution of a four-qubit isotropic Heisenberg XXX ring with periodic boundary conditions and a tunable next-nearest-neighbor exchange parameter \(\alpha\). Adopting a Bell-type product state as the initial condition, we obtain closed-form expressions for the square-root fidelity \(F(\rho(0),\rho(t))=|\cos(\phi/2)|\), the global \(l_1\)-norm coherence \(C_{l_1}(\rho(t))=\sin^2(\phi/2)\), and the bipartite entanglement of formation \(E_F(t)\) for the reduced pairs \((1,2)\) and \((3,4)\). These quantities are dictated entirely by the single phase \(\phi=(\alpha+1)t\). The fidelity shows oscillatory behavior with amplitude \(|\cos(\phi/2)|\) and reaches a quiescent dynamical regime at \(\alpha=-1\), characterized by \(F\equiv 1\) for all times. Meanwhile, the coherence follows \(C_{l_1}(\rho(t))=\sin^2(\phi/2)\), with a response sensitivity proportional to \(|\alpha+1|\), and fully extinguishes at the stabilization point \(\alpha=-1\). The entanglement of formation \(E_F(t)\), formulated as an entropic function of \(\phi\), undergoes striped periodic modulations and likewise freezes at the same parameter value. The unification of the phase ties together all observables, showing that larger \(|\alpha+1|\) drives faster evolution, with peak susceptibility occurring when \((\alpha+1)t=\pi/4+k\pi/2\). This complete analytical architecture supplies precise reference data suited for small-scale quantum hardware and charts a clear route for incorporating decoherence, thermal effects, and generalizations to larger spin assemblies.