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

TL;DR

This paper derives exact formulas for quantum resource dynamics in a four-qubit Heisenberg XXX chain with tunable interactions, revealing resource trade-offs, periodic behaviors, and regimes of frozen dynamics useful for benchmarking quantum devices.

Contribution

It provides closed-form expressions for fidelity, coherence, and entanglement evolution in a four-qubit chain with tunable coupling, unifying their behavior through a phase parameter.

Findings

Fidelity oscillates with phase, showing periodicity and frozen regimes at specific coupling values.

Coherence and entanglement exhibit banded oscillations and freezing at the special coupling point.

The phase parameter links the dynamics of all quantum resources, indicating maximal sensitivity at certain points.

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.