Inter-qubit correlation dynamics driven by mutual interactions

TL;DR

This work develops a correlation-tensor framework to study inter-qubit correlation dynamics under mutual two-body interactions, employing both two- and three-qubit systems. By deriving linear equations of motion for the correlation tensors and identifying characteristic frequencies from various Hamiltonians (XXX/XYZ Heisenberg, Dzyaloshinskii–Moriya, KSEA) under external fields, it characterizes the time evolution as trajectories on isopuric surfaces and delineates stationary correlation manifolds. For two qubits, the stationary states form small, well-defined regions (often tetrahedra) in parameter space, with explicit solutions in several models; extending to three qubits reveals analogous structures via sector-length dynamics $(A_1,A_2,A_3)$ that remain confined to isopuric constraints. A strong external field consistently stabilizes certain correlation characteristics, and the study highlights periodic and quasiperiodic behavior tied to frequency commensurability, offering a structured view of correlation dynamics relevant to spin-based quantum information tasks.

Abstract

A particularly useful tool for characterizing multi-qubit systems is the correlation tensor, providing an experimentally friendly and theoretically concise representation of quantum states. In this work, we analyze the evolution of the correlation tensor elements of quantum systems composed of $\frac12$-spins, generated by mutual interactions and the influence of the external field. We focus on two-body interactions in the form of anisotropic Heisenberg as well as antisymmetric exchange interaction models. The evolution of the system is visualized in the form of a trajectory in a suitable correlation space, which, depending on the system's frequencies, exhibits periodic or nonperiodic behavior. In the case of two $\frac12$-spins we study the stationary correlations for several classes of Hamiltonians, which allows a full characterization of the families of density matrices invariant under the evolution generated by the Hamiltonians. We discuss some common properties shared by the 2- and 3-qubit systems and show how a strong external field can play a stabilizing factor with respect to certain correlation characteristics.

Figures (6)

• Figure 1: Trajectories of time evolved correlation vector $(T_A,T_B, T_{AB})$ generated by the XYZ Heisenberg interactions model. The evolution takes place on the isopuric surface pertaining to the purity of the initial state $\textrm{Tr}\rho_{\textrm{rand}}^2 = 0.6416$. a) Closed trajectories generated by the isotropic Heisenberg Hamiltonian with $J_1=J_2=J_3$, for the initial state $\rho_{\textrm{rand}}$ (blue) and $\tilde{\rho}_{\textrm{rand}}$ (red) which is obtained from $\rho_{\textrm{rand}}$ by local change of basis. Both states $\rho_{\textrm{rand}}$ and $\tilde{\rho}_{\textrm{rand}}$ share a common initial correlation vector (black dot), whereas the time evolution follow different paths; b) Trajectories generated by the anisotropic Heisenberg XYZ interaction with incommensurable parameters $J_1=\sqrt{3}, J_2=\sqrt{2}, J_3=\sqrt{5}$ (in arbitrary units). A similar irregular quasiperiodic type of trajectory is obtained in the presence of arbitrary external field $\vec{B}$; c) Comparison of time evolved (with discrete time steps) correlation vector for the initial state represented by density matrix $\rho_{\textrm{rand}}$ (black) and the initial unphysical state $\rho^{\Gamma_B}_{\textrm{rand}}$ (blue and red).
• Figure 2: The evolution of the correlation vector generated by DM-type interactions with $D_1=D_2=D_3$ for different initial states in the correlation space (upper row) and the single qubit Bloch space (lower row). a) and b): Periodic evolution for random density matrix $\rho_{\textrm{rand}}$ in the absence of magnetic field. Note that the trajectory in the Bloch sphere does not intersect. A nonzero magnetic field $\vec{B}$ will make the corresponding trajectories non-periodic; c) and d): Periodic evolution for the initial state $|00\rangle$. $T^2_{AB}$ is upper-bounded by $\frac{17}{9}$. Note that the evolution in the Bloch sphere is uniplanar; e) and f): Quasiperiodic evolution for the initial state $|00\rangle$. A nonzero magnetic field $\vec{B}=(0,0,1)$ (in the units of $D$) renders $T^2_{AB}(t)$ to be a non-strictly-periodic function of $t$, the value of which can be arbitrarily close to 3.
• Figure 3: The evolution of the correlation vector generated by KSEA-type interactions with $K_1=K_2=K_3$ for different initial states in the correlation space (upper row) and the single qubit Bloch space (lower row). a) and b): Periodic evolution for random density matrix $\rho_{\textrm{rand}}$ in the absence of magnetic field. A nonzero magnetic field $\vec{B}$ makes the trajectories non-periodic; c) and d): Periodic evolution for the initial state $|00\rangle$. $T^2_{AB}$ is upper-bounded by $\frac{17}{9}$; e) and f): Quasiperiodic evolution for the initial state $|00\rangle$. A nonzero magnetic field $\vec{B}=(0,0,1)$ (in the units of $K$) renders $T^2_{AB}(t)$ to be a non-strictly-periodic function of $t$, the value of which is upper-bounded by $\approx 2.065$.
• Figure 4: Tetrahedra representing the range of parameters defining stationary states for different model Hamiltonians: a) stationary correlations \ref{['rostatani0']} for anisotropic Heisenberg Hamiltonian \ref{['hamani']} with zero magnetic field; b) stationary correlations \ref{['rostatani']} for anisotropic Heisenberg Hamiltonian \ref{['hamani']} with the magnetic field for $\Delta=1$; c) stationary correlations \ref{['rostatani']} for anisotropic Heisenberg Hamiltonian \ref{['hamani']} with the magnetic field for $\Delta=-0.5$; d) stationary correlations \ref{['rostatDM']} for Dzyaloshinskii-Moriya interaction model Hamiltonian \ref{['hamDM']} with an external magnetic field; e) stationary correlations \ref{['rostatKS']} for a model of KSEA interaction given by Hamiltonian \ref{['hamKS']} with an external magnetic field.
• Figure 5: The evolution of the correlation vector $(A_1,A_2,A_3)$ generated by different types of interactions for a random density matrix. The initial state $\rho_{\textrm{rand3}}$ is chosen as a proper 3:1 mixture of two randomly generated pure states, with purity $\textrm{Tr}\rho^2_{\textrm{rand3}}=0.7575$. a) A periodic trajectory in a form of a line for Heisenberg type interaction with $\vec{J}=(1,-1,0)$ in the presence of the external field $\vec{B}=(0,0,1)$; b) A squeezed trajectory for $\vec{J}=(1,-1,0)$ with a strong external directional field $\vec{B}=(0,0,5)$; c) A closed periodic trajectory for DM-type interactions $\vec{D}=(1,1,1)$ without the external field and d) with a strong external directional field $\vec{B}=(0,0,5)$; e) A closed periodic trajectory for KSEA-type interactions $\vec{K}=(1,1,1)$ without the external field; f) A squeezed trajectory for $\vec{K}=(1,1,1)$ with a strong external directional field $\vec{B}=(0,0,5)$.